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Chapter 4: Problem 50

Write each expression using only positive exponents. Assume that all variables represent nonzero real numbers. $$ (5 t)^{-3} $$

Short Answer

Expert verified
\( \frac{1}{125 t^3} \)

Step by step solution

01

Understanding Negative Exponents

A negative exponent indicates that the base should be moved to the denominator of a fraction, turning the exponent positive. For example, \text{ } \( a^{-n} = \frac{1}{a^n} \).
02

Apply the Negative Exponent Rule

Apply the rule \( a^{-n} = \frac{1}{a^n} \) to the given expression. Here, the base is \( 5t \) and the exponent is -3. Thus, it becomes \( \frac{1}{(5t)^3} \).
03

Simplify the Expression

Simplify the expression by cubing both 5 and t in the denominator: \( \frac{1}{(5t)^3} = \frac{1}{5^3 t^3} = \frac{1}{125 t^3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

negative exponents
The concept of negative exponents can be a bit tricky, but it is an essential part of algebra.
A negative exponent means that the base should be moved to the denominator, which in turn makes the exponent positive.
This can be summarized by the rule: \text{ } \( a^{-n} = \frac{1}{a^n} \).
With this rule, any base with a negative exponent can be rewritten as a fraction.
In our example, \((5t)^{-3}\) becomes \( \frac{1}{(5t)^3} \).
Don't forget, this works universally: whenever you see a negative exponent, think about flipping the base to make it positive.
exponent rules
Exponent rules are fundamental in simplifying expressions and solving equations.
Here are some important exponent rules:
  • Product of Powers: When multiplying with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).

  • Quotient of Powers: When dividing with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

  • Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).

  • Power of a Product: When raising a product to a power, apply the exponent to each factor: \( (ab)^n = a^n b^n \).

  • Power of a Quotient: When raising a quotient to a power, apply the exponent to both the numerator and denominator: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \).

These rules are crucial for working with exponents, as we saw in our example: we used the power of a product rule to cube both 5 and t: \((5t)^{-3} = \frac{1}{5^3 t^3} = \frac{1}{125 t^3}\).
simplifying expressions
Simplifying expressions is the process of reducing an expression to its most basic form.
The goal is to make expressions easier to work with and solve.
When simplifying:
  • Always apply exponent rules appropriately.

  • Combine like terms wherever possible.

  • Convert negative exponents using the rule: \( a^{-n} = \frac{1}{a^n} \).

    • Consider our example again: First, we had \((5 t)^{-3}\).
    To simplify, we converted the negative exponent into a positive one by moving the base to the denominator: \((5 t)^{-3} = \frac{1}{(5t)^3}\).
    Then, we simplified further by cubing both 5 and t: \( (5t)^3 = 5^3 t^3 \), resulting in \( \frac{1}{5^3 t^3} = \frac{1}{125 t^3} \).
    By following these steps consistently, we transform complex expressions into simpler, more manageable forms.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ a^{-3} a^{2} a^{-4} $$

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ \left(q-\frac{3}{4} r\right)^{2} $$

A student subtracted \(\left(15 x^{2}+12\right)-\left(7 x^{2}-3\right)\) vertically as follows. $$\begin{array}{r}15 x^{2}+12 \\\7 x^{2}-3 \\\\\hline 8 x^{2}+9\end{array}$$ Give the correct difference.

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{2^{2} y^{4}\left(y^{-3}\right)^{-1}}{2^{5} y^{-2}} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(x^{-3}\right)^{6} $$
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