Writing $\sqrt{y^3}$, $\sqrt[3]{x^2}$, and $\sqrt[4]{z^3}$ with Rational Exponents

Simplifying $9^{\frac{3}{2}}$, $125^{\frac{2}{3}}$, and $81^{\frac{3}{4}}$

Simplifying $16^{-\frac{3}{2}}$, $32^{-\frac{2}{5}}$, and $4^{-\frac{5}{2}}$

Simplifying $-25^{\frac{3}{2}}$, $-25^{-\frac{3}{2}}$, and $(-25)^{\frac{3}{2}}$

A technician uses a formula for pressure that includes the term p^(2/3). In this rational exponent, what is the mathematical role of the denominator 3?

A solar panel installer uses a formula for energy efficiency that includes the term E^(3/4). Match each part of this mathematical expression to its specific role when converting it into radical form.

A small business owner uses a formula to calculate the depreciation of equipment that includes the term v^(3/4). In this rational exponent, the number ____ is the index of the radical.

A logistics coordinator is manually simplifying the expression $25^{3/2}$ for a shipping volume calculation. According to the 'strategy tip' for rational exponents, the most efficient way to keep intermediate numbers small is to calculate $25^3$ first and then take the square root of the result.

Equivalent Radical Forms and the Power Property

A production planner is using a capacity-scaling formula that involves the expression $16^{3/4}$. Arrange the following steps in the correct order to evaluate this expression using the recommended strategy for keeping intermediate numbers as small as possible during manual calculation.

Defining Rational Exponents in Quality Assurance Documentation

Interpreting Technical Scaling Formulas

A technical writer is drafting a reference guide for internal mathematical formulas. To explain why the expression $a^{m/n}$ can be rewritten as $(a^{1/n})^m$, the writer needs to cite the foundational exponent rule used for this derivation. Which property should the writer cite?

A manufacturing technician is using a formula to calculate the expansion of a metal part, which includes the term $x^{5/2}$. When converting this expression into radical form to use on a standard calculator, which of the following is an equivalent representation of $x^{5/2}$?

Writing $(\sqrt[3]{2x})^4$ and $\sqrt{\left(\frac{3a}{4b}\right)^3}$ with Rational Exponents

Simplifying $27^{\frac{2}{3}}$ and $4^{\frac{3}{2}}$