Problem 65 - Algebra 2

Question

Simplify each expression. \(64^{\frac{2}{3}}\)

Step-by-Step Solution

Answer

Therefore, \(64^{\frac{2}{3}}\) equals 16.

Step 1: Identify the base, the root and the power

The base of the expression is 64, the root (denominator of the exponent) is 3 and the power (numerator of the exponent) is 2.

Step 2: Compute the cube root of the base

The cube root of 64 can be calculated as \(\sqrt[3]{64}\), which equals 4.

Step 3: Raise the result to the power of 2

After finding the cube root in the previous step, next is to raise this result to the power of 2, that is \(4^2\), which equals 16.

Key Concepts

Simplification of expressions

Simplification is an important skill in mathematics that allows us to make expressions more manageable and easier to understand. This concept especially applies when dealing with exponents, where expressions can sometimes look complex or daunting. By breaking these expressions down into smaller, simpler parts, we can better analyze and understand them.

In the expression given, such as \(64^{\frac{2}{3}}\), simplification involves understanding and manipulating the components smoothly. When simplifying similar expressions:

• Identify the base number (here, 64).
• Read the exponent; it consists of a numerator (2) and a denominator (3).
• Recognize rules like: any expression with a rational exponent represents a root, followed by a power (or vice versa).

By following these steps, complex expressions are transformed into more approachable, simple forms – facilitating easier calculations and efficiencies in solving.

Cube root

The cube root is the process of finding a number that, when multiplied by itself three times, results in the original number. In other words, it is the number whose third power is the original number. The cube root of a number \(a\) is written as \(\sqrt[3]{a}\).

For the example \(\sqrt[3]{64}\), you are finding which number, when multiplied by itself twice (4 in this case: 4 \(\times\) 4 \(\times\) 4), will give you 64. This number is the cube root. To compute:

• Think about common perfect cubes you remember, like 8, 27, 64, etc.
• You can memorize a few or use trial multiplication if unfamiliar.

Knowing the cube root helps handle expressions with rational exponents by breaking them down systematically and making calculation simpler.

Rational exponents

Rational exponents incorporate both roots and powers, representing a more comprehensive way to express numbers. An exponent that is a fraction, such as \(\frac{2}{3}\), indicates both steps combined into its process: to take a root and then apply a power.

Such expressions demand understanding the two-part process:

• The denominator, here 3, tells you which root to take (cube root).
• The numerator, here 2, signals the power to raise the result.

This means \(64^{\frac{2}{3}}\) means compute the cube root of 64 (4), then square the result (16).

Rational exponents are powerful because they allow calculations to remain compact and expressive, blending roots and powers seamlessly into one step. As you become familiar with this notation, complex expressions become simpler, offering enormous flexibility and clarity in mathematics.