Problem 95 - Algebra for College Students

Question

Use your calculator to evaluate each of the following. (a) \(16^{\frac{5}{2}}\) (b) \(25^{\frac{7}{2}}\) (c) \(16^{\frac{9}{4}}\) (d) \(27^{\frac{5}{3}}\) (e) \(343^{\frac{2}{3}}\) (f) \(512^{\frac{4}{3}}\)

Step-by-Step Solution

Answer

(a) 1024, (b) 78125, (c) 512, (d) 243, (e) 49, (f) 4096.

Step 1: Understanding the Exponent

For each expression, the exponent is a fraction, which means we deal with both roots and powers. For example, if the exponent is \( \frac{m}{n} \), we find the \(n\)th root of the base first, and then raise the result to the power of \(m\).

Step 2: Evaluating (a) \(16^{\frac{5}{2}}\)

1. The expression \(16^{\frac{5}{2}}\) can be rewritten as \((\sqrt{16})^5\) or \((16^{1/2})^5\).2. First, calculate the square root of 16: \(\sqrt{16} = 4\).3. Raise the result to the power of 5: \(4^5 = 1024\).4. Thus, \(16^{\frac{5}{2}} = 1024\).

Step 3: Evaluating (b) \(25^{\frac{7}{2}}\)

1. Rewrite \(25^{\frac{7}{2}}\) as \((\sqrt{25})^7\) or \((25^{1/2})^7\).2. Calculate the square root of 25: \(\sqrt{25} = 5\).3. Raise the result to the power of 7: \(5^7 = 78125\).4. Therefore, \(25^{\frac{7}{2}} = 78125\).

Step 4: Evaluating (c) \(16^{\frac{9}{4}}\)

1. Rewrite \(16^{\frac{9}{4}}\) as \((16^{1/4})^9\) or calculate \((\sqrt[4]{16})^9\).2. Calculate the fourth root of 16: \(\sqrt[4]{16} = 2\) because \(2^4 = 16\).3. Raise the result to the power of 9: \(2^9 = 512\).4. Hence, \(16^{\frac{9}{4}} = 512\).

Step 5: Evaluating (d) \(27^{\frac{5}{3}}\)

1. Rewrite \(27^{\frac{5}{3}}\) as \((27^{1/3})^5\) or \((\sqrt[3]{27})^5\).2. Calculate the cube root of 27: \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).3. Raise the result to the power of 5: \(3^5 = 243\).4. Thus, \(27^{\frac{5}{3}} = 243\).

Step 6: Evaluating (e) \(343^{\frac{2}{3}}\)

1. Rewrite \(343^{\frac{2}{3}}\) as \((343^{1/3})^2\) or \((\sqrt[3]{343})^2\).2. Calculate the cube root of 343: \(\sqrt[3]{343} = 7\) because \(7^3 = 343\).3. Raise the result to the power of 2: \(7^2 = 49\).4. Therefore, \(343^{\frac{2}{3}} = 49\).

Step 7: Evaluating (f) \(512^{\frac{4}{3}}\)

1. Rewrite \(512^{\frac{4}{3}}\) as \((512^{1/3})^4\) or \((\sqrt[3]{512})^4\).2. Calculate the cube root of 512: \(\sqrt[3]{512} = 8\) because \(8^3 = 512\).3. Raise the result to the power of 4: \(8^4 = 4096\).4. Thus, \(512^{\frac{4}{3}} = 4096\).

Key Concepts

Understanding Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. A fractional exponent also delivers a similar function, but with a twist. The fraction in the exponent, given as \( \frac{m}{n} \), signifies two actions. First, the base is subjected to an \( n \)-th root, which simplifies radical calculations. For instance, \( 16^{\frac{1}{2}} \) translates to \( \sqrt{16} \), which is 4. The second step involves taking this result and raising it to the power of \( m \). Consider \( 16^{\frac{5}{2}} \): reimagine it as \((\sqrt{16})^5\), which results in \( 1024 \). Understanding this key concept is vital for simplifying expressions that might otherwise appear complicated. Try breaking down the exponent step-by-step to ease the calculation process by handling one operation at a time.

Maximizing Calculator Use

Calculators are powerful tools in reducing complex calculations into simpler steps, especially with radical expressions and fractional exponents. Here’s how you can use a calculator effectively:

Firstly, rewrite the expression according to the fractional exponent, identifying both the root and power separately.

• For example, with \( 27^{\frac{5}{3}} \), determine that you need the cube root first, and then the fifth power.

Next, make use of the square root or radical functions readily available on most calculators to find the root.

• Calculate \( \sqrt[3]{27} = 3 \) first.

Finally, use the power or exponent function on your calculator to finalize the calculation.

• Enter 3 into your calculator, then use the power function to find \( 3^5 \), resulting in \( 243 \).

By breaking down the expression into these steps, you can approach even more daunting calculations with ease and accuracy.

Step-by-Step Evaluation Strategy

Breaking down your calculations into clear, manageable steps is crucial, especially when dealing with fractional exponents. Here’s a step-by-step strategy you can apply consistently:

**1. Rewrite:** Start by rewriting the expression using the fractional exponent \( \frac{m}{n} \), separating it into a root and a power.

• For instance, \( 343^{\frac{2}{3}} \) becomes \( (\sqrt[3]{343})^2 \).

**2. Find the Root:** Use the calculator or simple mental math if feasible to determine the root.

• Here, the cube root of 343 is 7, since \( 7^3 = 343 \).

**3. Apply the Power:** Finally, take the result and raise it to the given power.

• Since \( 7^2 = 49 \), the expression \( 343^{\frac{2}{3}} = 49 \).

Following this structured pathway helps to ensure no steps are skipped and that you consistently reach the correct solution. Remember, practice makes perfect, so try several exercises of varying difficulty to master the technique!