/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 77 Simplify. $$\left(a^{\frac{1}{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 77

Simplify. $$\left(a^{\frac{1}{8}}\right)^{\frac{2}{3}}$$

Short Answer

Expert verified
The simplified form is \a^{\frac{1}{12}}\.

Step by step solution

01

Understand Power of a Power Rule

Recall that the power of a power rule states \(\big(x^m\big)^n = x^{m \times n} \). This means that when raising a power to another power, you multiply the exponents.
02

Apply the Power of a Power Rule

Apply this rule to the given expression \(\big(a^{\frac{1}{8}}\big)^{\frac{2}{3}} = a^{\frac{1}{8} \times \frac{2}{3}}\).
03

Multiply the Exponents

Multiply the exponents: \(\frac{1}{8} \times \frac{2}{3} = \frac{1 \times 2}{8 \times 3} = \frac{2}{24} = \frac{1}{12}\). Thus, \(\big(a^{\frac{1}{8}}\big)^{\frac{2}{3}} = a^{\frac{1}{12}}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Power of a Power Rule
The power of a power rule is a fundamental concept in algebra. It tells us how to simplify expressions that involve an exponent raised to another exponent. When you see something like \(\big(x^m\big)^n\), you should remember that you can multiply the exponents together. So, \(\big(x^m\big)^n = x^{m \times n}\). This rule helps simplify complex expressions by reducing them to a single exponent. Keep in mind, this only works when the bases are the same. For example, in the given expression \(\big(a^{\frac{1}{8}}\big)^{\frac{2}{3}}\), the base is 'a', and we apply the power of a power rule by multiplying the given exponents: \( \frac{1}{8} \times \frac{2}{3}\).
Exponent Multiplication
Exponent multiplication is a key step when applying the power of a power rule. In this step, you multiply the two exponents together. For instance, given \(\big(a^{\frac{1}{8}}\big)^{\frac{2}{3}}\), begin by multiplying the two exponents. The multiplication looks like this: \(\frac{1}{8} \times \frac{2}{3}\). To carry this out:
  • Multiply the numerators: 1 \times 2 = 2
  • Multiply the denominators: 8 \times 3 = 24
Hence, the product of the exponents is \(\frac{2}{24} = \frac{1}{12}\). Now, you see that the original problem simplifies as: \(\big(a^{\frac{1}{8}}\big)^{\frac{2}{3}} = a^{\frac{1}{12}}\).
Algebraic Simplification
Algebraic simplification involves making an expression easier to understand or work with. Applying rules like the power of a power rule helps simplify expressions. In our example, we started with a complex expression \(\big(a^{\frac{1}{8}}\big)^{\frac{2}{3}}\). By applying the power of a power rule and performing exponent multiplication, we simplified it to \(a^{\frac{1}{12}}\). The final expression is simpler and more manageable. Simplification is not just about making expressions look better; it's also crucial for solving algebraic equations and for more advanced topics in mathematics. Always aim to express your final result in the simplest form possible.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Simplify. $$\frac{x^{\frac{7}{9}}}{x^{\frac{2}{9}}}$$

Rewrite the radical expression in exponential notation. $$\sqrt[5]{y}$$

Factor completely. $$2 x^{3}-7 x^{2}-4 x$$

In a study of broiler chickens, the daily water consumption per chicken, \(y\), depended on the age of the chicken in days, \(x\). A linear model of this relationship is $$ y=\left(\frac{5.28 \frac{\mathrm{mL}}{\text { chicken }}}{\text { age in days }}\right) x \text {. } $$ Find the percent increase in the amount of water needed by a chicken that is 20 days old compared to a chicken that is 12 days old. Round to the nearest percent.

Anesthesiologists use a cuffed endotracheal tube to intubate children. For children older than 1 year, a formula for finding the correct size of the internal diameter of the tube in millimeters, \(D\), is \(D=\frac{A}{4}+3.5\), where \(A\) is the age of the child in years. Predict the correct internal diameter of this tube for a 6-year-old child. (Source: Duracher et al., Paediatric Anaethesia, Feb. 2008)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.