Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. (5t)^-3

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Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$, where $a$ is a nonzero number and $n$ is a positive integer.

Apply the negative exponent rule to the expression $(5t)^{-3}$ by rewriting it as $\frac{1}{(5t)^3}$.

Expand the denominator by raising both 5 and $t$ to the power of 3: $(5t)^3 = 5^3 \cdot t^3$.

Write the expression as $\frac{1}{5^3 \cdot t^3}$ to eliminate the negative exponent.

Evaluate \$5^3$ as $5 \times 5 \times 5$, but leave the expression in this form if you are not asked for a numerical value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a⁻³ equals 1 divided by a³. This rule allows rewriting expressions without negative exponents by moving factors between numerator and denominator.

Exponentiation of Products

When raising a product to a power, each factor inside the parentheses is raised to that power separately. For instance, (5t)⁻³ equals 5⁻³ times t⁻³. This property helps simplify expressions involving powers of products.

Evaluating Expressions with Variables

When variables represent nonzero real numbers, expressions can be simplified by applying exponent rules without concern for division by zero. This assumption ensures that rewriting negative exponents as reciprocals is valid and the expression can be evaluated if numerical values are given.

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