Problem 557 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(16^{\frac{3}{2}}\) (c) \(64^{\frac{4}{3}}\) (b) \(125^{\frac{5}{3}}\)

Step-by-Step Solution

Answer

For (a) \(16^{\frac{3}{2}}=64\), (c) \(64^{\frac{4}{3}}=256\), and (b) \(125^{\frac{5}{3}}=3125\).

Step 1: Simplify \(16^{\frac{3}{2}}\)

First, recognize that \(16\) is a power of \(2\). Specifically, \(16 = 2^4\). Rewrite \(16^{\frac{3}{2}}\) as \((2^4)^{\frac{3}{2}}\). According to the properties of exponents, \((a^m)^n = a^{m \cdot n}\). Therefore, \((2^4)^{\frac{3}{2}} = 2^{4 \cdot \frac{3}{2}} = 2^6\). Since \(2^6 = 64\), we have \(16^{\frac{3}{2}} = 64\).

Step 2: Simplify \(64^{\frac{4}{3}}\)

Recognize that \(64\) is a power of \(4\). Specifically, \(64 = 4^3\). Rewrite \(64^{\frac{4}{3}}\) as \((4^3)^{\frac{4}{3}}\). Using the properties of exponents, \((a^m)^n = a^{m \cdot n}\). Thus, \((4^3)^{\frac{4}{3}} = 4^{3 \cdot \frac{4}{3}} = 4^4\). Since \(4^4 = 256\), we have \(64^{\frac{4}{3}} = 256\).

Step 3: Simplify \(125^{\frac{5}{3}}\)

Identify that \(125\) is a power of \(5\). Specifically, \(125 = 5^3\). Rewrite \(125^{\frac{5}{3}}\) as \((5^3)^{\frac{5}{3}}\). Using the properties of exponents, \((a^m)^n = a^{m \cdot n}\). Therefore, \((5^3)^{\frac{5}{3}} = 5^{3 \cdot \frac{5}{3}} = 5^5\). Since \(5^5 = 3125\), we have \(125^{\frac{5}{3}} = 3125\).

Key Concepts

Exponential Notation

Exponential notation is a way to express repeated multiplication of the same number. It is written as: \(a^n\), where 'a' is the base and 'n' is the exponent. For example, \(2^3\) means 2 multiplied by itself three times: \(2 \times 2 \times 2 = 8\). This notation makes it easier to handle large or complex numbers.
Understanding exponential notation is the first step in simplifying exponents.
It is essential for mastering more advanced concepts in algebra and calculus.

Properties of Exponents

The properties of exponents are rules that simplify expressions involving exponents. Here are the key properties you need to know:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)

For example, in step 1 of the solution, we used the power of a power property to simplify \( (2^4)^{\frac{3}{2}} = 2^{4 \times \frac{3}{2}}\). Proper application of these properties makes complex problems more manageable.

Simplification Process

The simplification process involves applying the properties of exponents to rewrite expressions in their simplest form. Here is a step-by-step outline:

1. Identify the base and rewrite it as a power if possible (e.g., 16 = \(2^4\)).
2. Apply the power of a power property: \( (a^m)^n = a^{m \times n} \).
3. Calculate the new exponent and simplify further if needed.

For example, in step 2, 64 was recognized as \(4^3\), making it easier to simplify \(64^{\frac{4}{3}}\) into \(4^{4} = 256\). This methodical approach ensures precision and clarity in solving exponent problems.