Problem 55 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \left(4 x^{3}\right)^{-2} $$

Step-by-Step Solution

Answer

\((\frac{1}{16x^{6}})\)

Step 1: Apply the power of a power rule

First, we will handle the inner part of the parentheses. The power of a power rule says that to raise a power to a power, multiply the exponents. So we have \( (4x^3)^{-2}= 4^{-2} x^{3*(-2)}\).

Step 2: Simplify the numerical part

Next, we simplify \(4^{-2}\) which means finding the reciprocal of \(4^2\). That gives us \((\frac{1}{4^2})\). So now we have, \((\frac{1}{4^2})x^{3*(-2)} = (\frac{1}{16})x^{3*(-2)}\).

Step 3: Apply the rule of negative exponents

Now we apply the rule of negative exponents to \(x^{3*(-2)}\), giving us \((\frac{1}{x^{3*2}})\). Hence we have \((\frac{1}{16})(\frac{1}{x^{3*2}})\).

Step 4: Multiply the fractions

Finally, to simplify further, multiply the two fractions together, giving \((\frac{1}{16x^{3*2}})\).

Key Concepts

Power of a Power Rule

Understanding the power of a power rule is crucial when simplifying complex exponential expressions. In algebra, this rule comes into play when you have an exponent raised to another exponent, as seen with expressions like \( (a^m)^n \). To simplify, you simply multiply the exponents, resulting in \( a^{m*n} \).

For example, in the given exercise, we see \( (4x^3)^{-2} \). Applying the power of a power rule here involves multiplying the exponent of \( x^3 \) by -2, transforming the expression into \( 4^{-2} \times x^{3*(-2)} \) or more clearly, \( 4^{-2} \times x^{-6}\). Keep in mind that this rule works for any real numbers \( a, m, \) and \( n\), as long as \( a \) is not zero.

When simplifying these types of problems, remember that you’re only multiplying the exponents, not the bases, unless they are the same, in which case you would add them (another rule for another time). This simplification cuts down on the complexity and gets us one step closer to our final, simplified expression.

Negative Exponents

The concept of negative exponents often puzzles students but is quite simple once you understand the underlying principle. A negative exponent tells us to take the reciprocal of the base raised to the absolute value of the exponent. For example, \( a^{-n} \) is the same as \( \frac{1}{a^n} \) where \( n \) is a positive integer.

In our exercise \( 4^{-2} \times x^{-6} \) from the previous section, we simplified \( 4^{-2} \) by finding its reciprocal, \( \frac{1}{4^2} \) which simplifies to \( \frac{1}{16}\). We similarly treat \( x^{-6} \) as \( \frac{1}{x^6} \) and incorporate this into our expression.

Thus, the rule of negative exponents flips the position of the base—taking it from the numerator to the denominator, or vice versa, ultimately simplifying the expression. This rule helps avoid dealing with negative numbers in exponents by transforming them into a more familiar form—positive exponents.

Simplify Fractions in Algebra

Simplifying fractions is a foundational skill in algebra, and it's important to manage both numerical and variable parts of a fraction efficiently. To simplify fractions, find any common factors in the numerator and the denominator and divide them out. When dealing with algebraic expressions, you'll also need to apply exponent rules, as we've seen earlier.

In our exercise, after applying the power of a power and negative exponent rules, we arrived at \( (\frac{1}{16})(\frac{1}{x^{6}})\). Multiplying the two fractions together, as they both have a denominator of 1, yields \( \frac{1}{16x^6}\).

At this stage, there are no common factors in the numerator and the denominator, so the fraction is already simplified. Remember, when simplifying fractions with variables, only cancel factors that are common to the numerator and denominator and apply the exponent rules correctly to ensure the most simplified form of the algebraic fraction is obtained.