When evaluating an exponential expression with a numerical base, the base is multiplied by itself the number of times indicated by the exponent. For example, to evaluate $$2^3$$, the base $$2$$ is multiplied by itself three times ($$2 \cdot 2 \cdot 2$$) to give $$8$$. Similarly, evaluating $$3^2$$ involves multiplying the base $$3$$ by itself two times ($$3 \cdot 3$$) to yield $$9$$. This demonstrates how exponential notation is expanded to calculate a final numerical value.

Claude

In an exponential expression, the **exponent** is the superscript number that indicates how many times the base is used as a factor in repeated multiplication. It dictates the total number of times the base must be multiplied by itself. For instance, in the expression $$2^3$$, the number $$3$$ is the exponent, which tells us that the base $$2$$ undergoes repeated multiplication three times: $$2 \cdot 2 \cdot 2$$.

Algebraic Exponent

https://openstax.org/details/books/intermediate-algebra-2e

OpenStax Intermediate Algebra

The order of operations is a set of established mathematical guidelines that ensure a given expression consistently produces the identical result for everyone, preventing different interpretations of the same problem. To apply the order of operations, proceed through these four steps sequentially: first, simplify all expressions within parentheses or other grouping symbols, starting with the innermost ones and working outward; second, evaluate any terms with exponents; third, carry out all multiplication and division operations as they appear from left to right, since they possess equal priority; and fourth, perform all addition and subtraction operations from left to right, as they also share equal priority.

Order of Operations in Algebra

To evaluate the polynomial $$2x^2 + 3x + 8$$ when $$x = 4$$, one must substitute $$4$$ for every occurrence of $$x$$ to obtain $$2(4)^2 + 3(4) + 8$$. Following the order of operations, the exponent is evaluated first, producing $$2(16) + 3(4) + 8$$. Next, the multiplication steps yield $$32 + 12 + 8$$. Adding the terms sequentially results in $$44 + 8$$, which mathematically simplifies to a final value of $$52$$. This step-by-step example demonstrates the proper application of substitution and the order of operations in evaluating an algebraic expression.

Evaluating $$2x^2 + 3x + 8$$ at $$x = 4$$

When $$x = 3$$, evaluating these three expressions involves substituting $$3$$ for $$x$$ and simplifying. For $$x^2$$, the substitution becomes $$3^2$$, which equals $$9$$. In the case of $$4^x$$, the variable serves as an exponent, so replacing $$x$$ yields $$4^3$$, which involves multiplying $$4$$ by itself three times to reach $$64$$. For the multi-term expression $$3x^2 + 4x + 1$$, inserting $$3$$ results in $$3(3)^2 + 4(3) + 1$$. Applying the order of operations, one must evaluate the exponent first to obtain $$3(9) + 4(3) + 1$$, then perform multiplications to reach $$27 + 12 + 1$$, and finally add the components together to yield $$40$$.

Evaluating $$x^2$$, $$4^x$$, and $$3x^2 + 4x + 1$$ at $$x = 3$$

Simplify the expressions $$(-3)^4$$ and $$-3^4$$ to understand the impact of parentheses on the base and the final sign when raising a negative number to an even power:

**ⓐ $$(-3)^4 = 81$$:** The base is $$-3$$. The exponent instructs to multiply $$-3$$ by itself four times: $$(-3)(-3)(-3)(-3) = 81$$. Multiplying an even number of negative factors yields a positive product.

**ⓑ $$-3^4 = -81$$:** The base is $$3$$. The expression indicates the opposite of $$3^4$$. First, calculate the power: $$3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$$. Next, apply the negative sign to obtain $$-81$$.

These examples highlight that parentheses include the negative sign in the repeatedly multiplied base, producing a positive result for an even exponent. Without parentheses, the exponent is applied only to the positive integer, and the result is negated afterward according to the order of operations.

Learn Before

Related