Problem 76 - Elementary and Intermediate Algebra

Question

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. \(5^{-\sqrt{5}}\)

Step-by-Step Solution

Answer

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

Step 1: 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, \((2^{\sqrt{3}})^{\sqrt{3}} = 2^{\sqrt{3} \cdot \sqrt{3}} = 2^3 = 8\).

Step 2: 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 \(7^{\sqrt{3} + 2\sqrt{3}} = 7^{3\sqrt{3}}\).

Step 3: 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 \(5^{2\sqrt{2}}\).

Step 4: 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, \(5^{-\sqrt{5}} = \frac{1}{5^{\sqrt{5}}}\).

Key Concepts

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’s 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’s 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.