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Chapter 10: Problem 84

Use the laws of exponents to simplify. Do not use negative exponents in any answers. $$ \left(5^{5 / 4}\right)^{3 / 7} $$

Short Answer

Expert verified
The simplified expression is \ 5^{15 / 28}. \

Step by step solution

01

Understand the Power of a Power Rule

The power of a power rule states that \[ \left(a^{m}\right)^{n} = a^{m \times n}. \] In other words, you can multiply the exponents together.
02

Identify the Exponents

In the expression \[ \left(5^{5 / 4}\right)^{3 / 7} \], \ a = 5, \ m = \frac{5}{4} \ and \ n = \frac{3}{7} \.
03

Multiply the Exponents Together

Using the power of a power rule, multiply the exponents \[ \frac{5}{4} \times \frac{3}{7} = \frac{5 \times 3}{4 \times 7} = \frac{15}{28} \].
04

Simplify the Expression

Replace the exponents with the result obtained from step 3. The simplified expression is \[ 5^{15 / 28}. \]

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Key Concepts

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

Power of a Power Rule
When dealing with exponents, one of the critical rules to remember is the power of a power rule. This rule helps simplify expressions where an exponent is raised to another exponent. The power of a power rule states that: \[ \left(a^{m}\right)^{n} = a^{m \times n} \].
Multiplying Exponents
In the expression \( \left(5^{5 / 4}\right)^{3 / 7} \), the exponents need to be multiplied together to simplify according to the power of a power rule. \(a = 5\), \(m = \frac{5}{4}\), and \(n = \frac{3}{7}\) in this example. To do this:
  • Multiply the numerators from both exponents: \(5 \times 3 = 15\)
  • Multiply the denominators from both exponents: \(4 \times 7 = 28\)
Combining these results, you get \( \frac{15}{28} \). This means that the expression simplifies to \(5^{15/28}\).
Simplifying Expressions
Simplifying expressions involving exponents often involves using exponent rules like the power of a power rule. Here, we used: \[ \left(5^{5 / 4}\right)^{3 / 7} \]
Applying the power of a power rule: \[ \left(5^{5 / 4}\right)^{3 / 7} = 5^{\left(5/4 \times 3/7\right)} = 5^{15/28} \].
The result is the simplified form of the original expression.

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Most popular questions from this chapter

Review simplifying rational expressions (Sections\(7.1-7.5)\) Perform the indicated operation and, if possible, simplify. $$ \frac{a^{-1}+b^{-1}}{a b}[7.5] $$

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