Question:
Simplify $$9\sqrt{10} + 4\sqrt{10}$$. What happens to the coefficients, and what happens to the radical part?

Sample answer:
Add the coefficients: $$9 + 4 = 13$$. The radical part stays the same. So $$9\sqrt{10} + 4\sqrt{10} = 13\sqrt{10}$$.

Key points:
- Like square roots have the same radicand.
- Add or subtract only the coefficients.
- The radical part stays unchanged.
- $$9\sqrt{10} + 4\sqrt{10} = 13\sqrt{10}$$

Rubric:
Full credit for stating that coefficients are added while the radical part stays unchanged, with the correct simplified result. Partial credit if only one part is stated. No credit if the radicands are combined.

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Like square roots are added and subtracted using the same procedure as **combining like terms**: add or subtract the coefficients while keeping the radical part unchanged. Just as $$3x + 8x = 11x$$ (the coefficients $$3$$ and $$8$$ are added while the variable $$x$$ stays the same), the expression $$3\sqrt{x} + 8\sqrt{x} = 11\sqrt{x}$$ is simplified by adding the coefficients $$3$$ and $$8$$ while the radical $$\sqrt{x}$$ remains unchanged. Similarly, $$\sqrt{3} + \sqrt{3} = 2\sqrt{3}$$, because a single $$\sqrt{3}$$ has an implicit coefficient of $$1$$, and $$1 + 1 = 2$$.

Square roots with **different** radicands cannot be combined. The expression $$\sqrt{2} + \sqrt{7}$$ must be left as is — it is a common error to incorrectly merge the two radicals into $$\sqrt{2 + 7}$$. In fact, $$\sqrt{2} + \sqrt{7} \neq \sqrt{9}$$, because the radical sign acts as a grouping symbol and only governs the expression directly underneath it.

Adding and Subtracting Like Square Roots

To simplify $$2\sqrt{2} - 7\sqrt{2}$$, recognize that both terms share the identical radicand $$2$$ and are therefore like square roots. You can combine them by subtracting their coefficients, exactly as you would with algebraic like terms. Compute $$2 - 7 = -5$$, and keep the radical unchanged to get $$-5\sqrt{2}$$. This illustrates how subtracting a larger coefficient from a smaller one results in a negative coefficient, just as $$2x - 7x = -5x$$.

Simplifying $$2\sqrt{2} - 7\sqrt{2}$$

Simplify the expression $$3\sqrt{y} + 4\sqrt{y}$$, which involves adding two like square roots whose radicand is a variable.

Both terms contain $$\sqrt{y}$$, so the radicals are like. Add the coefficients while keeping the radical part unchanged: $$3 + 4 = 7$$.

$$3\sqrt{y} + 4\sqrt{y} = 7\sqrt{y}$$

This example shows that the procedure for combining like square roots works identically whether the radicand is a number or a variable. As long as the expressions under the radical signs match, the coefficients are simply added together.

Simplifying $$3\sqrt{y} + 4\sqrt{y}$$

Simplify the expression $$4\sqrt{x} - 2\sqrt{y}$$, which contains two square roots with **different** radicands.

The first term has radicand $$x$$ and the second has radicand $$y$$. Because the radicands are not the same, these are **not** like square roots. Their coefficients cannot be combined — the expression is already in its simplest form.

$$4\sqrt{x} - 2\sqrt{y} = 4\sqrt{x} - 2\sqrt{y}$$

This is analogous to the algebraic expression $$4x - 2y$$, which cannot be simplified further because $$x$$ and $$y$$ are different variables. Attempting to subtract unlike square roots would be an error, just as attempting to combine unlike terms would be.

Simplifying $$4\sqrt{x} - 2\sqrt{y}$$

Simplify the expression $$5\sqrt{13} + 4\sqrt{13} + 2\sqrt{13}$$, which involves adding three like square roots.

All three terms contain the radical $$\sqrt{13}$$, so they are all like square roots. Add the three coefficients while keeping the radical part unchanged: $$5 + 4 + 2 = 11$$.

$$5\sqrt{13} + 4\sqrt{13} + 2\sqrt{13} = 11\sqrt{13}$$

This example extends the technique to more than two terms. Just as combining three like variable terms such as $$5x + 4x + 2x = 11x$$ works by adding all the coefficients at once, the same approach applies when the common factor is a square root rather than a variable.

Simplifying $$5\sqrt{13} + 4\sqrt{13} + 2\sqrt{13}$$

Simplify the expression $$2\sqrt{6} - 6\sqrt{6} + 3\sqrt{3}$$, which contains a mixture of like and unlike square roots.

The first two terms, $$2\sqrt{6}$$ and $$-6\sqrt{6}$$, share the radicand $$6$$ and are therefore like square roots. The third term, $$3\sqrt{3}$$, has a different radicand ($$3$$) and cannot be combined with the others. Subtract the coefficients of the like radicals: $$2 - 6 = -4$$.

$$2\sqrt{6} - 6\sqrt{6} + 3\sqrt{3} = -4\sqrt{6} + 3\sqrt{3}$$

This example illustrates that when an expression contains both like and unlike square roots, only the like ones are combined. The unlike term is carried through unchanged — exactly as one would do with a mixed expression such as $$2a - 6a + 3b = -4a + 3b$$ in algebra.

Simplifying $$2\sqrt{6} - 6\sqrt{6} + 3\sqrt{3}$$

To simplify $$2\sqrt{5n} - 6\sqrt{5n} + 4\sqrt{5n}$$, observe that all three terms are like square roots because they share the identical radicand $$5n$$. You can combine them by summing their coefficients: $$2 - 6 + 4 = 0$$. Multiplying this zero coefficient by the radical yields $$0$$. Thus, the entire expression simplifies to $$0$$. This is directly comparable to the algebraic simplification of like terms where $$2x - 6x + 4x$$ results in $$0x$$, which simplifies to $$0$$.

Simplifying $$2\sqrt{5n} - 6\sqrt{5n} + 4\sqrt{5n}$$

Simplify the expression $$\sqrt{20} + 3\sqrt{5}$$, where one of the square roots must be simplified before the terms can be combined.

The two terms do not initially appear to be like radicals because their radicands ($$20$$ and $$5$$) are different. Simplify $$\sqrt{20}$$ by extracting its largest perfect square factor:

$$\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}$$

Now rewrite the original expression with the simplified radical:

$$2\sqrt{5} + 3\sqrt{5}$$

Both terms contain $$\sqrt{5}$$, so they are like radicals. Combine the coefficients: $$2 + 3 = 5$$.

$$\sqrt{20} + 3\sqrt{5} = 5\sqrt{5}$$

This example illustrates that a radical expression which appears to contain unlike terms may actually contain like radicals once each square root has been simplified.

Simplifying $$\sqrt{20} + 3\sqrt{5}$$

Simplify the expression $$\sqrt{48} - \sqrt{75}$$, where both square roots must be simplified before the terms can be combined.

Neither $$\sqrt{48}$$ nor $$\sqrt{75}$$ is in simplified form. Extract the largest perfect square factor from each radicand:

$$\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$$

$$\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}$$

Rewrite the expression with the simplified radicals:

$$4\sqrt{3} - 5\sqrt{3}$$

Both terms now contain $$\sqrt{3}$$, making them like radicals. Subtract the coefficients: $$4 - 5 = -1$$.

$$\sqrt{48} - \sqrt{75} = -\sqrt{3}$$

This example shows that when both radicals need simplification, simplifying each one separately can reveal a common radicand. The result is negative because the coefficient of the subtracted term ($$5$$) exceeds the coefficient of the first term ($$4$$).

Simplifying $$\sqrt{48} - \sqrt{75}$$

Simplify the expression $$\frac{3}{4}\sqrt{192} - \frac{5}{6}\sqrt{108}$$, where both terms have fractional coefficients and the radicals must be simplified.

Simplify each radical by extracting the largest perfect square factor:

$$\frac{3}{4}\sqrt{192} = \frac{3}{4} \cdot \sqrt{64} \cdot \sqrt{3} = \frac{3}{4} \cdot 8 \cdot \sqrt{3} = 6\sqrt{3}$$

$$\frac{5}{6}\sqrt{108} = \frac{5}{6} \cdot \sqrt{36} \cdot \sqrt{3} = \frac{5}{6} \cdot 6 \cdot \sqrt{3} = 5\sqrt{3}$$

In each term, the fractional coefficient is multiplied by the integer from the simplified radical, producing a whole-number coefficient.

Rewrite the expression:

$$6\sqrt{3} - 5\sqrt{3}$$

Both terms contain $$\sqrt{3}$$, so combine the coefficients: $$6 - 5 = 1$$.

$$\frac{3}{4}\sqrt{192} - \frac{5}{6}\sqrt{108} = \sqrt{3}$$

This example extends the technique to fractional coefficients. When the integer from the simplified radical multiplies with the fraction, the result may simplify to a whole number — here $$\frac{3}{4} \cdot 8 = 6$$ and $$\frac{5}{6} \cdot 6 = 5$$ — making the final combination straightforward.

Simplifying $$\frac{3}{4}\sqrt{192} - \frac{5}{6}\sqrt{108}$$

Simplify two expressions involving higher roots that are already like radicals.

**ⓐ $$\sqrt[3]{4x} + \sqrt[3]{4x} = 2\sqrt[3]{4x}$$:**

Both terms are cube roots with the same radicand $$4x$$, so they are like radicals. Each term has an implicit coefficient of $$1$$. Add the coefficients: $$1 + 1 = 2$$.

$$\sqrt[3]{4x} + \sqrt[3]{4x} = 2\sqrt[3]{4x}$$

**ⓑ $$4\sqrt[4]{8} - 2\sqrt[4]{8} = 2\sqrt[4]{8}$$:**

Both terms are fourth roots with the same radicand $$8$$, so they are like radicals. Subtract the coefficients: $$4 - 2 = 2$$.

$$4\sqrt[4]{8} - 2\sqrt[4]{8} = 2\sqrt[4]{8}$$

In both parts, the radicals share the same index and radicand, so only the coefficients are combined while the radical factor remains unchanged — the same approach used when combining like square roots or like terms in algebra.

Simplifying $$\sqrt[3]{4x} + \sqrt[3]{4x}$$ and $$4\sqrt[4]{8} - 2\sqrt[4]{8}$$

Simplify two expressions where the higher roots must first be simplified to reveal like radicals that can be combined.

**ⓐ $$\sqrt[3]{54} - \sqrt[3]{16} = \sqrt[3]{2}$$:**

Neither radicand is in simplified form. Rewrite each using its largest perfect cube factor: $$54 = 27 \cdot 2$$ and $$16 = 8 \cdot 2$$. Apply the Product Property of cube roots:

$$\sqrt[3]{27} \cdot \sqrt[3]{2} - \sqrt[3]{8} \cdot \sqrt[3]{2}$$

Since $$\sqrt[3]{27} = 3$$ and $$\sqrt[3]{8} = 2$$:

$$3\sqrt[3]{2} - 2\sqrt[3]{2}$$

Both terms now contain $$\sqrt[3]{2}$$, so they are like radicals. Combine the coefficients: $$3 - 2 = 1$$.

$$\sqrt[3]{54} - \sqrt[3]{16} = \sqrt[3]{2}$$

**ⓑ $$\sqrt[4]{48} + \sqrt[4]{243} = 5\sqrt[4]{3}$$:**

Rewrite each radicand using its largest perfect fourth power factor: $$48 = 16 \cdot 3$$ and $$243 = 81 \cdot 3$$. Apply the Product Property of fourth roots:

$$\sqrt[4]{16} \cdot \sqrt[4]{3} + \sqrt[4]{81} \cdot \sqrt[4]{3}$$

Since $$\sqrt[4]{16} = 2$$ and $$\sqrt[4]{81} = 3$$:

$$2\sqrt[4]{3} + 3\sqrt[4]{3}$$

Both terms contain $$\sqrt[4]{3}$$, making them like radicals. Combine the coefficients: $$2 + 3 = 5$$.

$$\sqrt[4]{48} + \sqrt[4]{243} = 5\sqrt[4]{3}$$

In both parts, the original radicands appear different, but simplifying each radical by extracting the largest perfect $$n$$th power factor reveals a common radical that allows the terms to be combined. This extends the simplify-then-combine strategy from square roots to higher-order roots.

Simplifying $$\sqrt[3]{54} - \sqrt[3]{16}$$ and $$\sqrt[4]{48} + \sqrt[4]{243}$$

Simplify the expression $$\sqrt[4]{3xy} + 5\sqrt[4]{3xy} - 4\sqrt[4]{3xy}$$, noticing that each term is a fourth root with the exact same multi-variable radicand, $$3xy$$. These are like radicals, and the first term carries an implicit coefficient of $$1$$. Combining the coefficients ($$1 + 5 - 4 = 2$$) yields the simplified expression $$2\sqrt[4]{3xy}$$. This example highlights that standard rules for combining like radicals work smoothly even with multiple variables, provided the root index and all parts of the radicand are perfectly identical across the terms.

Simplifying $$\sqrt[4]{3xy} + 5\sqrt[4]{3xy} - 4\sqrt[4]{3xy}$$

To add or subtract higher-order roots, apply the same logic used for combining like terms or like square roots: add or subtract the numerical coefficients while leaving the radical factor untouched. This operation requires that the terms be like radicals, meaning they must have identical indices and identical radicands. For instance, $$9\sqrt[4]{42x} - 2\sqrt[4]{42x} = 7\sqrt[4]{42x}$$ because both terms are fourth roots with the radicand $$42x$$. If the radicals do not match initially, you should first simplify each one by extracting the largest perfect $$n$$th power factor. Often, this simplification process will reveal hidden like radicals that can then be successfully combined.

Adding and Subtracting Like Radicals

When adding or subtracting square roots, the terms may not initially appear to be like radicals because their radicands differ. However, after **simplifying** each square root by removing the largest perfect-square factor, the simplified forms may reveal like radicals that can then be combined.

The general strategy has two stages:
1. **Simplify each radical** by factoring out the largest perfect square from the radicand and applying the Product Property.
2. **Combine like radicals** by adding or subtracting their coefficients, just as one combines like terms in algebra.

When a radical already has a numerical coefficient — for example, $$ 5\sqrt{18} $$ — simplifying the radical produces a product of three factors (such as $$ 5 \cdot 3 \cdot \sqrt{2} $$). The Associative Property of Multiplication allows these numerical factors to be multiplied together first, yielding a single coefficient in front of the radical ($$ 15\sqrt{2} $$). This mirrors the algebraic simplification $$ 5(3x) = 15x $$, where the two numerical factors are combined while the variable part remains unchanged.

Adding and Subtracting Square Roots that Need Simplification

Simplify the expression $$ 5\sqrt{18} - 2\sqrt{8} $$, where both terms already have numerical coefficients and the radicals must also be simplified.

Simplify each radical by extracting the largest perfect square factor:
$$ 5\sqrt{18} = 5 \cdot \sqrt{9} \cdot \sqrt{2} = 5 \cdot 3 \cdot \sqrt{2} = 15\sqrt{2} $$
$$ 2\sqrt{8} = 2 \cdot \sqrt{4} \cdot \sqrt{2} = 2 \cdot 2 \cdot \sqrt{2} = 4\sqrt{2} $$

In each case, the Associative Property of Multiplication allows the original coefficient and the integer from the simplified radical to be multiplied together into a single new coefficient.

Rewrite the expression:
$$ 15\sqrt{2} - 4\sqrt{2} $$

Both terms contain $$ \sqrt{2} $$, so combine the coefficients: $$ 15 - 4 = 11 $$.

$$ 5\sqrt{18} - 2\sqrt{8} = 11\sqrt{2} $$

This example demonstrates that when each term already carries a coefficient, the simplification process produces a product of three factors — the original coefficient, the integer from the simplified radical, and the remaining radical — which are then consolidated using the Associative Property before combining like radicals.

Simplifying $$ 5\sqrt{18} - 2\sqrt{8} $$

Which pair of square roots are 'like' and can be combined by adding?

Match each square root expression with its simplified result.

Simplify: $$8\sqrt{5} + 2\sqrt{5} = $$ _____

Arrange the steps in order to simplify $$9\sqrt{5} + \sqrt{5}$$.

In $$\sqrt{15} + \sqrt{15} = 2\sqrt{15}$$, what coefficient does $$\sqrt{15}$$ have when no number is written in front of it?

Case context:
$$5\sqrt{7}$$ and $$3\sqrt{7}$$ are like square roots because they share the same radicand, $$7$$.

Question:
Simplify $$5\sqrt{7} + 3\sqrt{7}$$. Explain what happens to the coefficients and what happens to the radicand.

Sample answer:
$$5\sqrt{7} + 3\sqrt{7} = 8\sqrt{7}$$. Like square roots are combined like like terms: add the coefficients, and leave the radicand unchanged.

Key points:
- $$5\sqrt{7}$$ and $$3\sqrt{7}$$ are like square roots since both have radicand $$7$$.
- Add the coefficients: $$5 + 3 = 8$$.
- The radicand $$7$$ stays the same and is never added.
- The simplified answer is $$8\sqrt{7}$$.

Rubric:
Full credit: states the correct sum $$8\sqrt{7}$$ and explains that only the coefficients are added while the radicand stays the same. Partial credit: correct sum without a clear explanation, or a clear explanation with an arithmetic error. No credit: radicand is changed or added.

Simplify $$5\sqrt{7} + 3\sqrt{7}$$. Explain what happens to the coefficients and what happens to the radicand.

Question:
What makes two square roots "like," and how do you add or subtract them?

Sample answer:
Two square roots are "like" when they have the same radicand (the number under the radical sign). To add or subtract like square roots, add or subtract the coefficients and keep the radicand unchanged: $$3\sqrt{x} + 8\sqrt{x} = 11\sqrt{x}$$. A square root with no visible coefficient has an implicit coefficient of $$1$$, so $$\sqrt{3} + \sqrt{3} = 2\sqrt{3}$$. If the radicands are different, the square roots cannot be combined: $$\sqrt{2} + \sqrt{7}$$ must stay as is, since $$\sqrt{2} + \sqrt{7} \neq \sqrt{9}$$.

Key points:
- Like square roots have the same radicand.
- Only the coefficients are added or subtracted.
- The radicand does not change.
- A square root with no coefficient has an implicit coefficient of 1.
- Square roots with different radicands cannot be combined.

Rubric:
Full credit: states that like square roots share the same radicand, explains that only coefficients are added or subtracted while the radicand stays the same, and notes that different radicands cannot be combined. Partial credit: covers some but not all of these points. No credit: incorrect or missing explanation.

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