Problem 47 - Algebra 1: Concepts and Skills

Question

In Exercises 47 and \(48,\) find and correct the error. $$ \begin{aligned} \left(\frac{6 a}{b^{2}}\right)^{3} &=\frac{6 a^{3}}{b^{2} \cdot 3} \\ &=\frac{6 a^{3}}{b^{8}} \end{aligned} $$

Step-by-Step Solution

Answer

Following the correct laws of exponentiation, the expression \(\left(\frac{6 a}{b^{2}}\right)^{3}\) simplifies to \(\frac{216a^3}{b^6}\), not \(\frac{6a^3}{b^8}\), as originally stated.

Step 1: Mind the Exponentiation Law

Firstly, remember the law of exponentiation: \((a/b)^n = a^n / b^n\). This law implies that when a fraction is raised to an exponent, both the numerator (top) and the denominator (bottom) are raised to that power.

Step 2: Apply the Correct Exponentiation Law

Apply the correct exponentiation law to the given expression \(\left(\frac{6 a}{b^{2}}\right)^{3}\). This gives \(\frac{(6a)^3}{(b^2)^3}\). Note that the exponent 3 applies to all terms in the numerator and the denominator separately.

Step 3: Simplify the Expression Further

Simplify the expression further. The term \((6a)^3\) simplifies to \(216a^3\), and \((b^2)^3\) simplifies to \(b^6\). Therefore, the entire expression simplifies to \(\frac{216a^3}{b^6}\).

Key Concepts

Exponentiation Law

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. In the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. This operation tells us to multiply the base by itself \(n\) times.

When it comes to fractions, the exponentiation law is crucial. According to the law:

• \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)

This means that both the numerator and the denominator are raised to the power \(n\).

For instance, in the expression \(\left(\frac{6a}{b^2}\right)^3\), every part of the fraction needs to be raised to the third power separately. This results in \((6a)^3\) over \((b^2)^3\). Understanding this law helps prevent mistakes in algebraic manipulations and ensures predictions of increasing or decreasing expressions are accurate.

Simplifying Expressions

Simplifying an expression means reducing it to its simplest form while maintaining equality. It's akin to clearing up clutter from your desk, leaving only what's essential and neat.

When you are simplifying an expression involving exponentiation and fractions, break it into steps. Using the example of the fraction \(\frac{(6a)^3}{(b^2)^3}\), apply exponent rules to each part:

• First, simplify \((6a)^3\). This means multiplying \(6a\) by itself three times: \(6a \times 6a \times 6a\).
• Calculate each step: \(6^3 = 216\), since \(6 \times 6 \times 6 = 216\), and \(a^3 = a \times a \times a\).

Likewise, for the denominator \((b^2)^3\), raise \(b^2\) to the third power:

• \(b^2 \times b^2 \times b^2 = b^{2+2+2} = b^6\).

Thus, the original complex expression simplifies beautifully to \(\frac{216a^3}{b^6}\). Knowing how to simplify expressions not only helps to solve problems faster but also enhances understanding of mathematical relationships.

Fractions in Algebra

Fractions in algebra can often seem intimidating, but with a little practice, they become manageable. Essentially, they are very similar to fractions in basic arithmetic but incorporate variables instead of just numbers.

When manipulating algebraic fractions, remember:

• The rules for operations (addition, subtraction, multiplication, division) apply just like they do with numerical fractions.
• Be careful with the placement of terms; remember every factor in the denominator and numerator counts.

In the exercise given, applying exponentiation to fractions highlights another important skill set: raising the entire fraction \(\left(\frac{6a}{b^2}\right)^3\) properly balances powers in both numerator and denominator. This balance is crucial to maintaining equivalences. After raising terms correctly, make sure to simplify to see clearer results—a simpler look at truth is always beneficial. Practicing these steps in algebraic fractions bolsters confidence and fluency in more complex algebraic operations.