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Chapter 11: Problem 76

Simplify each expression. Write answers using positive exponents. a. \(\left(2^{\sqrt{3}}\right)^{\sqrt{3}}\) b. \(7^{\sqrt{3}} 7^{\sqrt{12}}\) c. \(\frac{5^{6 \sqrt{2}}}{5^{4 \sqrt{2}}}\) d. \(5^{-\sqrt{5}}\)

Short Answer

Expert verified
a. 8, b. \(7^{3\sqrt{3}}\), c. \(5^{2\sqrt{2}}\), d. \(\frac{1}{5^{\sqrt{5}}}\)

Step by step solution

01

Simplify the first expression

The expression is \( (2^{\sqrt{3}})^{\sqrt{3}} \). To simplify, use the power of a power property: \((a^m)^n = a^{m \cdot n}\). So, \((2^{\sqrt{3}})^{\sqrt{3}} = 2^{\sqrt{3} \cdot \sqrt{3}} = 2^3 = 8\).
02

Simplify the second expression

The expression is \( 7^{\sqrt{3}} \times 7^{\sqrt{12}} \). Use the product of powers property: \(a^m \cdot a^n = a^{m+n}\). Combine the exponents: \(\sqrt{3} + \sqrt{12}\). Recognize \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\). This gives \(7^{\sqrt{3} + 2\sqrt{3}} = 7^{3\sqrt{3}}\).
03

Simplify the third expression

The expression is \( \frac{5^{6 \sqrt{2}}}{5^{4 \sqrt{2}}} \). Use the quotient of powers property: \(\frac{a^m}{a^n} = a^{m-n}\). Subtract the exponents: \(6\sqrt{2} - 4\sqrt{2} = 2\sqrt{2}\). The simplified form is \(5^{2\sqrt{2}}\).
04

Simplify the fourth expression

The expression is \(5^{-\sqrt{5}}\). To express it with a positive exponent, use the property \(a^{-n} = \frac{1}{a^n}\). Thus, \(5^{-\sqrt{5}} = \frac{1}{5^{\sqrt{5}}}\).

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Key Concepts

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

Power of a Power Property
When you see an expression raised to another power, like \((a^m)^n\), you use the power of a power property to simplify. This rule tells us that you multiply the exponents: \((a^m)^n = a^{m \cdot n}\). This works because you're essentially considering \(a^m\) repeated \(n\) times.

This concept is useful to simplify expressions like \((2^{\sqrt{3}})^{\sqrt{3}}\). Here, you multiply \(\sqrt{3} \times \sqrt{3}\), which gives \(\sqrt{3}^2\), and since \(\sqrt{3}^2 = 3\), the expression becomes \(2^3 = 8\). It鈥檚 a straightforward process once you recognize how to multiply the exponents.
Product of Powers Property
You often encounter expressions where the same base is multiplied, such as \(a^m \cdot a^n\). In these situations, use the product of powers property: simply add the exponents: \(a^m \cdot a^n = a^{m+n}\).

For example, simplifying \(7^{\sqrt{3}} \times 7^{\sqrt{12}}\) requires adding the exponents. Recognizing \(\sqrt{12} = 2\sqrt{3}\) helps. Then, \(\sqrt{3} + 2\sqrt{3} = 3\sqrt{3}\), so you get \(7^{3\sqrt{3}}\). It鈥檚 all about identifying parts and adding the exponents properly.
Quotient of Powers Property
When you divide expressions with the same base, like \(\frac{a^m}{a^n}\), apply the quotient of powers property: subtract the exponents: \(a^{m-n}\). This subtraction works because dividing is the reverse of multiplying, which corresponds to exponents subtraction.

Consider \(\frac{5^{6 \sqrt{2}}}{5^{4 \sqrt{2}}}\). By subtracting the exponents, \(6\sqrt{2} - 4\sqrt{2}\), we find \(2\sqrt{2}\). Thus, the expression simplifies to \(5^{2\sqrt{2}}\). Keep the same base and rely on subtraction for simplification.
Negative Exponents
Expressions with negative exponents represent the inverse, or reciprocal. For any base \(a\), the rule \(a^{-n} = \frac{1}{a^n}\) applies. This turns the expression into its reciprocal form, making the exponent positive.

Look at \(5^{-\sqrt{5}}\). By using the property, it becomes \(\frac{1}{5^{\sqrt{5}}}\). Transform any negative exponent using this reciprocal approach, which makes equations easier to handle without negative powers.

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Most popular questions from this chapter

Use a calculator to evaluate each expression, if possible. Express all answers to four decimal places. See Using Your Calculator: Evaluating Base-e (Natural) Logarithms. $$\ln 1.72$$

Hydrogen lon Concentration. Find the hydrogen ion concentration of a saturated solution of calcium hydroxide whose \(\mathrm{pH}\) is 13.2

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \log (x-90)+\log x=3 $$

Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=1.001 $$

Use a graphing calculator to solve each equation. If an answer is not exact, round to the nearest tenth. See Using Your Calculator: Solving Exponential Equations Graphically or Solving Logarithmic Equations Graphically. $$ \ln (2 x+5)-\ln 3=\ln (x-1) $$
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