Problem 232 - Intermediate Algebra

Question

In the following exercises, evaluate. \(8 u^{2} v^{3}\) when \(u=-\frac{3}{4}\) and \(v=-\frac{1}{2}\)

Step-by-Step Solution

Answer

- \frac{9}{16}

Step 1 - Substitute the Values

Substitute the given values of \(u\) and \(v\) into the expression. Therefore, \(u = -\frac{3}{4}\) and \(v = -\frac{1}{2}\).

Step 2 - Compute \(u^2\)

First, find the square of \(u\). Substituting, \(u^2 = \left(-\frac{3}{4}\right)^2 = \frac{9}{16}\).

Step 3 - Compute \(v^3\)

Next, find the cube of \(v\). Substituting, \(v^3 = \left(-\frac{1}{2}\right)^3 = -\frac{1}{8}\).

Step 4 - Multiply Components Together

Now, multiply the components together using the expression: \[ 8 \cdot u^2 \cdot v^3 = 8 \cdot \frac{9}{16} \cdot -\frac{1}{8}\] This simplifies to: \[ 8 \cdot \frac{9}{16} \cdot -\frac{1}{8} = 8 \cdot 9 \cdot -1 \div (16 \cdot 8) = -\frac{9}{16}\].

Key Concepts

substitution

To evaluate an algebraic expression like the one given, it's crucial to understand the concept of substitution. Substitution simply means replacing variables in the expression with their respective values.
For instance, in the exercise, you substitute the values given for variables u and v directly into the expression.
Replace u with \(-\frac{3}{4}\) and v with \(-\frac{1}{2}\), and enter these values into the equation \(8 u^{2} v^{3}\).
This step is foundational as it sets the stage for further calculations.

exponentiation

Exponentiation is the process of raising a number to a power.
In this exercise, you need to calculate \(u^2\) and \(v^3\).
Let's break this down:
First, find the square of u.
When u is \(-\frac{3}{4}\), you square it by multiplying \(-\frac{3}{4}\) by itself:
\[\left(-\frac{3}{4}\right)^2 = \frac{9}{16}\].
Squaring a negative number yields a positive result.
Next, calculate the cube of v.
With v as \(-\frac{1}{2}\), you multiply it by itself three times:
\[-\left(-\frac{1}{2}\right)^3 = -\frac{1}{8}\].
Note that cubing a negative number results in a negative value.
Understanding how to handle exponents helps simplify the expression step-by-step.

multiplication of fractions

After computing the exponentiations, you will need to multiply the resulting fractions to evaluate the expression.
Specifically, you will multiply \(8 \cdot \frac{9}{16} \cdot -\frac{1}{8}\).
Multiplication of fractions involves multiplying the numerators with each other and the denominators with each other.
Here's how it's done:
Multiply the numerators: 8 \cdot 9 \cdot -1 = -72.
Multiply the denominators: 1 \cdot 16 \cdot 8 = 128.
Therefore, the result is \(-\frac{72}{128}\).
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, 8:
\[\frac{-72 \div 8}{128 \div 8} = -\frac{9}{16}\].
Understanding each step in the multiplication and simplification process is essential for correctly evaluating the expression.