Problem 14 - Intermediate Algebra

Question

For Problems \(1-30\), evaluate each numerical expression. $$ 64^{\frac{2}{3}} $$

Step-by-Step Solution

Answer

The answer is 16.

Step 1: Understanding the Problem

The expression we need to evaluate is \(64^{\frac{2}{3}}\). This involves employing both exponent rules and knowledge of roots.

Step 2: Rewrite using Root and Exponent

We can rewrite the expression \(64^{\frac{2}{3}}\) as \((64^{\frac{1}{3}})^2\). This means first taking the cube root of 64 and then squaring the result.

Step 3: Calculate the Cube Root of 64

The cube root of 64, denoted as \(64^{\frac{1}{3}}\), is the number that, when multiplied by itself three times, gives 64. Since \(4 \times 4 \times 4 = 64\), the cube root of 64 is 4.

Step 4: Square the Result

Now that we have \((64^{\frac{1}{3}}) = 4\), we need to square it. Squaring 4 means calculating \(4^2\), which is equal to 16.

Key Concepts

Exponent Rules

When working with exponent rules, it's important to know how to break down complex expressions into simpler parts. Exponents represent repeated multiplication. For example, in the expression \(a^m\), the base \(a\) is multiplied by itself \(m\) times.
Some essential exponent rules include:

• Product of Powers Rule: When multiplying with the same base, add the exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Negative Exponent Rule: A negative exponent indicates a reciprocal: \(a^{-m} = 1/a^m\).

Understanding these rules helps simplify calculations, especially when dealing with fractional exponents, like in our problem \(64^{\frac{2}{3}}\). Here, the rule for fractional exponents applies, meaning the expression can be split into a root and an additional power.

Cube Roots

Cube roots are the inverse operation of cubing a number. When you seek the cube root of a number, you are searching for a number that when used three times in multiplication results in the original number. For example, the cube root of 64 means finding a number \(x\) such that \(x \times x \times x = 64\).
To easily find cube roots, knowing some perfect cubes is handy, such as:

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)

For the given problem, the cube root \(64^{\frac{1}{3}}\) simplifies to 4, as \(4 \times 4 \times 4 = 64\). By correctly identifying and calculating cube roots, you can simplify expressions involving fractional exponents more easily.

Numerical Expressions

Numerical expressions are mathematical expressions involving numbers and operations. In this context, they include operations like addition, subtraction, multiplication, division, and exponentiation. Proper evaluation of numerical expressions requires an understanding of order of operations and simplifying complex expressions.
One such order of operations rule is PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This rule guides the systematic and correct evaluation of expressions.
In the exercise \(64^{\frac{2}{3}}\), understanding the operations embedded within the exponent is crucial:

• First, solve the root \(64^{\frac{1}{3}} = 4\).
• Then, apply the exponent \(4^2 = 16\).

Breaking down expressions into simpler steps and understanding their components helps in achieving accurate results efficiently.