Problem 76 - 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 Expression a

The expression is \( (2^{\sqrt{3}})^{\sqrt{3}} \). According to the power of a power rule \( (a^m)^n = a^{m \cdot n} \), we multiply the exponents: \[ \left(2^{\sqrt{3}}\right)^{\sqrt{3}} = 2^{\sqrt{3} \cdot \sqrt{3}} = 2^{3} \].After simplification, the expression becomes \( 8 \).

Step 2: Simplify Expression b

The expression is \( 7^{\sqrt{3}} 7^{\sqrt{12}} \). By using the rule \( a^m \cdot a^n = a^{m+n} \), the exponents are added:\[ 7^{\sqrt{3}} \cdot 7^{\sqrt{12}} = 7^{\sqrt{3} + \sqrt{12}} \].Calculate \( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \) and add them: \[ \sqrt{3} + 2\sqrt{3} = 3\sqrt{3} \].Thus, the expression simplifies to \( 7^{3\sqrt{3}} \).

Step 3: Simplify Expression c

The expression is \( \frac{5^{6 \sqrt{2}}}{5^{4 \sqrt{2}}} \). Using the quotient of powers rule \( \frac{a^m}{a^n} = a^{m-n} \), subtract the exponents:\[ \frac{5^{6 \sqrt{2}}}{5^{4 \sqrt{2}}} = 5^{6\sqrt{2} - 4\sqrt{2}} = 5^{2\sqrt{2}} \].The simplified expression is \( 5^{2\sqrt{2}} \).

Step 4: Express d with Positive Exponents

The expression is \( 5^{-\sqrt{5}} \). To express this using a positive exponent, recall that \( a^{-m} = \frac{1}{a^m} \). Thus:\[ 5^{-\sqrt{5}} = \frac{1}{5^{\sqrt{5}}} \].

Key Concepts

Power of a Power Rule

When dealing with exponentiation, you'll often encounter situations where an exponent is raised by another exponent. This is where the Power of a Power Rule comes to the rescue. This rule states that

• \((a^m)^n = a^{m \cdot n}\), which helps greatly in simplifying complex exponential expressions.

For instance, in the exercise:

• \((2^{\sqrt{3}})^{\sqrt{3}}\) is simplified using this rule.
• You simply multiply the exponents: \(\sqrt{3} \cdot \sqrt{3}\), which equals \(3\).
• Thus, it simplifies to \(2^3 = 8\).

This rule is crucial for breaking down expressions where an exponent is further powered, making your calculations smoother and your homework easier!

Quotient of Powers Rule

The Quotient of Powers Rule is another handy tool when simplifying expressions involving exponents. This rule tells us that:

• \(\frac{a^m}{a^n} = a^{m-n}\). This essentially means you subtract the exponent in the denominator from the exponent in the numerator.

Let's see how this was applied:

• In the example \(\frac{5^{6\sqrt{2}}}{5^{4\sqrt{2}}}\), you apply the rule by subtracting the exponents:
• \(5^{6\sqrt{2} - 4\sqrt{2}}\) simplifies to \(5^{2\sqrt{2}}\).

Subtracting exponents is particularly useful for simplifying fractional expressions, especially when dealing with large or complex exponent values.

Rational Exponents

Rational exponents are exponents that are fractions or contain roots. They are a sleek way of expressing roots with exponents, where:

• \(a^{m/n}\) represents the \(n\)-th root of \(a\) raised to the power of \(m\),

allowing you to combine roots and powers in a single expression. In the given problems:

• The expression \(7^{\sqrt{12}}\) involved a square root under the exponent, which can be re-expressed for simplification.
• Calculating or simplifying these requires recognizing how fractional or irrational numbers combine under exponentiation rules.

Understanding how to manipulate these exponents is vital for worksheets and exercises, especially those involving more advanced algebra.

Simplifying Expressions

Simplifying expressions is an essential part of algebra that involves making expressions easier to work with while retaining their values.

• It means breaking down or combining like terms to achieve more straightforward terms.

In this context, one of the examples is converting negative exponents to positive ones. For instance:

• The expression \(5^{-\sqrt{5}}\) can be simplified by recalling that a negative exponent signifies a reciprocal.
• Thus, it becomes \(\frac{1}{5^{\sqrt{5}}}\), replacing the negative with a fractional format that is easier to interpret.

These techniques in simplification transform complex problems into manageable solutions, ensuring you understand the core principles of each problem without unnecessary clutter.