Problem 11 - Beginning and Intermediate Algebra

Question

Evaluate. \(\left(\frac{2}{3}\right)^{4}\)

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{16}{81} \).

Step 1: Understand the Expression

The expression \( \left(\frac{2}{3}\right)^4 \) means you need to multiply \( \frac{2}{3} \) by itself four times. In other words, it is \( \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \).

Step 2: Apply the Properties of Exponents

Using the property of exponents for fractions, \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \), apply it to the given fraction: \( \left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4} \).

Step 3: Calculate the Numerator

Find \( 2^4 \), which means multiplying 2 by itself 4 times: \( 2 \times 2 \times 2 \times 2 = 16 \).

Step 4: Calculate the Denominator

Find \( 3^4 \), which means multiplying 3 by itself 4 times: \( 3 \times 3 \times 3 \times 3 = 81 \).

Step 5: Combine the Results

Combine your results from Steps 3 and 4 to get the value of the original expression: \( \frac{16}{81} \).

Key Concepts

Fractional Exponents

When dealing with fractional exponents, it's important to recognize that these represent both roots and powers. A fractional exponent like \( x^{1/n} \) is essentially the \( n \)-th root of \( x \). In simpler terms, \( x^{1/2} \) is the square root of \( x \), \( x^{1/3} \) is the cube root, and so on.

One key point about fractional exponents is that they create a special relationship between multiplication and roots:

• When the numerator is one, the exponent is a pure root.
• If the numerator is not one, say \( x^{m/n} \), it means you take the \( n \)-th root of \( x^m \). So, \( x^{2/3} \) means \( (x^2)^{1/3} \).

Understanding these basics allows you to manipulate these forms for simplification or solving equations. Remember, fractional exponents help streamline expressions and expand your algebraic toolbox.

Properties of Exponents

The properties of exponents are rules that help you manipulate expressions involving powers. Understanding these properties can simplify complex problems:

• Product of Powers: \( a^m \times a^n = a^{m+n} \). Multiply and add exponents when bases are the same.
• Power of a Power: \( (a^m)^n = a^{m \times n} \). Multiply exponents when raising a power to another power.
• Power of a Product: \( (ab)^n = a^n \times b^n \). Distribute the exponent to each base inside the brackets.
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \). Subtract exponents when dividing with the same base.
• Zero Exponent: \( a^0 = 1 \). Any non-zero base to the zero power is one.
• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \). A negative exponent indicates a reciprocal.

Applying these principles simplifies expressions considerably and aids in solving equations more efficiently. In the original equation \( \left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4} \), we used the power of a quotient property, allowing us to handle both the numerator and denominator separately.

Fraction Multiplication

Multiplying fractions is a straightforward process. Unlike addition or subtraction, you don't need common denominators. You simply multiply the numerators together and the denominators together. This can be described in steps:

• Multiply the numerators: Top numbers multiply with top numbers.
• Multiply the denominators: Bottom numbers multiply with bottom numbers.
• Simplify if possible: Always check if the result can be simplified by finding common factors.

For example, if you're calculating \( \frac{2}{3} \times \frac{4}{5} \), you'd perform the operation as follows:

• Numerators: \( 2 \times 4 = 8 \)
• Denominators: \( 3 \times 5 = 15 \)
• Result: \( \frac{8}{15} \)

This makes evaluating expressions with powers like \( \left(\frac{2}{3}\right)^4 \) more manageable, as it boils down to multiplying \( \frac{2}{3} \) by itself repeatedly.