Problem 12 - Elementary Algebra

Question

For Problems \(1-20\), find the value of each numerical expression. For example, \(2^{4}=2 \cdot 2 \cdot 2 \cdot 2=16\). $$ \left(\frac{3}{4}\right)^{3} $$

Step-by-Step Solution

Answer

The value of the expression is \( \frac{27}{64} \).

Step 1: Understand the Expression

We are given the expression \( \left( \frac{3}{4} \right)^3 \). This means that we need to multiply the fraction \( \frac{3}{4} \) by itself a total of three times.

Step 2: Multiply the Fractions

Multiply the fraction by itself three times: \[ \left( \frac{3}{4} \right) \times \left( \frac{3}{4} \right) \times \left( \frac{3}{4} \right) \] First, multiply the numerators: \[ 3 \times 3 \times 3 = 27 \] Then, multiply the denominators: \[ 4 \times 4 \times 4 = 64 \]

Step 3: Write the Result

After multiplying the numerators and denominators, we obtain the fraction: \[ \frac{27}{64} \] This is the simplified form of the expression since 27 and 64 have no common factors.

Key Concepts

Understanding Numerical Expressions

Numerical expressions represent one or more numbers and operations gathered together. It's like a recipe that tells you which mathematical operations to perform and in what sequence. These operations can include addition, subtraction, multiplication, division, and even exponents. When dealing with exponents, there is an extra step involved because you repeat the multiplication. For instance, in the expression \( 2^4 \), the number 2 is multiplied by itself 4 times, resulting in 16. Understanding how to handle these operations is key to solving any numerical expression accurately. Efficiency comes when you master how each operation affects the others, especially in combination with fractions and exponents. Each step in a numerical expression serves as a building block towards the final outcome.

Breaking Down Fractions

Fractions are a way of expressing numbers that aren't whole. They consist of a numerator and a denominator. The numerator indicates how many parts you take; the denominator tells how many parts make up a whole. For example, \( \frac{3}{4} \) means 3 parts out of 4 equal parts, or three-quarters of something. It's crucial to remember that both the numerator and the denominator help us grasp the exact size or quantity represented. Consider fractions like pieces of a pie; whether you're eating a quarter or three-quarters, the fraction helps you know exactly how much you have. Understanding fractions ensures you accurately perform operations like addition, subtraction, or raising them to a power, as seen in the original solution. Most importantly, with operations involving fractions, always check whether the results can be simplified.

Multiplying Fractions

When multiplying fractions, the process is straightforward compared to adding or subtracting them. You multiply the numerators together to get the new numerator and the denominators together for the new denominator. Suppose you have three fractions to multiply: \( \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \). Start by multiplying the numerators: - \( 3 \times 3 \times 3 = 27 \)Next, multiply the denominators: - \( 4 \times 4 \times 4 = 64 \)The result is \( \frac{27}{64} \). Notice that multiplying fractions does not require finding a common denominator, simplifying the process. After obtaining a fraction result, always check if it can be reduced to its simplest form. For \( \frac{27}{64} \), no further simplification is needed. This approach ensures accuracy and simplicity in dealing with fractional multiplication.