Problem 6 - College Algebra with Modeling and Visualization

Question

Simplify the expression without a calculator $$ -4(8)^{-2 / 3} $$

Step-by-Step Solution

Answer

-1

Step 1: Understand the Expression

We are given the expression \(-4(8)^{-2/3}\). This is a combination of a constant multiplier and a base with a negative rational exponent. Our task is to simplify this expression.

Step 2: Simplify the Exponent

We start by simplifying \((8)^{-2/3}\). A negative exponent indicates that we take the reciprocal of the base. Hence, \((8)^{-2/3} = \frac{1}{(8)^{2/3}}\).

Step 3: Compute the Cube Root and Square

Next, we need to interpret \((8)^{2/3}\). The exponent \(2/3\) tells us to first take the cube root of 8, and then square the result. The cube root of 8 is 2, so we have \((8)^{1/3} = 2\). Now square this result: \(2^2 = 4\). Therefore, \((8)^{2/3} = 4\).

Step 4: Combine Steps for Reciprocal

Now, substitute the result from Step 3 into the reciprocal form: \((8)^{-2/3} = \frac{1}{4}\).

Step 5: Multiply by the Constant

Finally, multiply this result by \(-4\) as indicated in the original expression: \(-4 \times \frac{1}{4} = -1\).

Key Concepts

Rational Exponents

Rational exponents, often referred to as fractional exponents, are a way to express powers and roots in one notation. They combine the concept of the power of a number with its root. The general form is: \( a^{m/n} \), where \( a \) is a base, \( m \) is the numerator, which represents the power, and \( n \), the denominator, indicates the root. For example, \( 8^{2/3} \) means you first need to take the cube root of 8, which is 2, and then raise this result to the power of 2, giving you 4. Rational exponents simplify expressions that involve both roots and powers, making calculations more streamlined.

• Numerator (m): Power of the base.
• Denominator (n): Root of the base.

When you see a fractional exponent, it can be tackled in two primary steps: calculating the root and then applying the power. This two-step approach helps break down complex operations into manageable parts.

Simplification Process

The simplification process in algebra involves converting expressions into their simplest form. This makes expressions easier to evaluate or compare with others. In this exercise, simplifying \( -4(8)^{-2/3} \) involves several steps.First, acknowledge the negative exponent, which flips the fraction. This means \((8)^{-2/3} = \frac{1}{(8)^{2/3}}\).

• The cube root \((8)^{1/3}\), which results in 2.
• Next, square the result: \(2^2 = 4\).
• This changes \((8)^{2/3}\) into 4.

So, \((8)^{-2/3}\) is transformed into \(\frac{1}{4}\), which is much easier to work with.Finally, multiply by the constant, -4, to reach the solution of -1. The key in simplification lies in systematically tackling each component, breaking down expressions to their fundamental forms for easy manipulation.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. They frequently appear in algebra, especially in expressions requiring simplification. For example, in the expression \( (8)^{-2/3} \), the negative exponent suggests reversing the base, thus creating the reciprocal.

• Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)
• Inverts the base: Transformations go from \( a \) to \( \frac{1}{a} \)

In practice, this means when you see a negative sign with an exponent, think of flipping the number to a fraction. In exercises like the given one, handle the exponent first, simplifying the power in the denominator, and then work up to the entire expression.Understanding this helps with any algebra problem featuring negative exponents, allowing you to rewrite and simplify expressions efficiently.