Chapter 5: Problem 25

Express the following using single powers: (a) \(\frac{4^{1 / 3} 4^{8}}{4^{-1 / 2}}\) (b) \(\sqrt{y} \sqrt[3]{y} \sqrt[4]{y}\)

Short Answer

Expert verified

Question: Simplify the expressions into their simplest form: (a) \(\frac{4^{1 / 3} 4^{8}}{4^{-1 / 2}}\) (b) \(\sqrt{y} \sqrt[3]{y} \sqrt[4]{y}\) Answer: (a) \(4^{102/6}\) (b) \(y^{13/12}\)

Step by step solution

(Step 1: Apply exponent properties in expression part (a))

For part (a), we have the expression \(\frac{4^{1 / 3} 4^{8}}{4^{-1 / 2}}\). Recall that if we have powers with the same base and we would like to multiply or divide them, we can use exponent properties: 1. Multiply powers with the same base: \(a^{m} \times a^{n} = a^{m+n}\) 2. Divide powers with the same base: \(\frac{a^{m}}{a^{n}} = a^{m-n}\) Let's apply these properties to our expression: \(\frac{4^{1/3} 4^8}{4^{-1/2}} = 4^{1/3} 4^8 4^{1/2} = 4^{1/3 + 8 + 1/2}\)

(Step 2: Simplify exponent in expression part (a))

Next, we need to simplify the exponent inside the parentheses: \(1/3 + 8 + 1/2 = 1/3 + 24/3 +1/2 = \frac{25}{3} + \frac{26}{6} = \frac{50}{6} + \frac{52}{6} = \frac{102}{6}\) So our simplified expression for part (a) is: \(4^{102/6}\)

(Step 3: Write the expression part (b) in terms of powers)

For part (b), we have the expression \(\sqrt{y} \sqrt[3]{y} \sqrt[4]{y}\). First, let's rewrite this expression using fractional exponents: \(y^{1/2} y^{1/3} y^{1/4}\)

(Step 4: Apply exponent properties in expression part (b))

As we did in part (a), we can use exponent properties to combine these powers of y: \(y^{1/2} y^{1/3} y^{1/4} = y^{1/2+1/3+1/4}\)

(Step 5: Simplify exponent in expression part (b))

Finally, we need to simplify the exponent inside the parentheses: \(1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12 = 13/12\) So our simplified expression for part (b) is: \(y^{13/12}\) Now we have expressed both expressions in terms of single powers: (a) \(4^{102/6}\) (b) \(y^{13/12}\)

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Express the following using single powers: (a) \(\frac{4^{1 / 3} 4^{8}}{4^{-1 / 2}}\) (b) \(\sqrt{y} \sqrt[3]{y} \sqrt[4]{y}\)

Short Answer

Step by step solution

(Step 1: Apply exponent properties in expression part (a))

(Step 2: Simplify exponent in expression part (a))

(Step 3: Write the expression part (b) in terms of powers)

(Step 4: Apply exponent properties in expression part (b))

(Step 5: Simplify exponent in expression part (b))

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Linear Momentum

Electric Charge Field and Potential

Atoms and Radioactivity

Modern Physics

Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.