Problem 112 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ -\left(\frac{b^{8}}{625}\right)^{3 / 4} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{b^6}{125}\).

Step 1: Identify Base and Exponent

The expression we need to simplify is \(-\left(\frac{b^{8}}{625}\right)^{3/4}\). Here, the base is \(\frac{b^{8}}{625}\) and the exponent is \(\frac{3}{4}\).

Step 2: Apply the Power Rule for Fractions

When a fraction is raised to an exponent, we raise both the numerator and denominator to that exponent separately. Therefore, we have: \[ -\left(\frac{b^8}{625}\right)^{3/4} = -\frac{(b^8)^{3/4}}{(625)^{3/4}} \]

Step 3: Simplify the Numerator

Simplify \((b^8)^{3/4}\) using the power of a power property \((x^m)^n = x^{mn}\): \[ (b^8)^{3/4} = b^{8 \cdot \frac{3}{4}} = b^{6} \]

Step 4: Simplify the Denominator

Since \(625 = 5^4\), simplify \((625)^{3/4}\) as follows: \[ (625)^{3/4} = (5^4)^{3/4} = 5^{4 \cdot \frac{3}{4}} = 5^3 = 125 \]

Step 5: Combine and Finalize Expression

Substitute the simplified results back into the fraction: \[ -\frac{b^6}{125} \] This is the simplified form of the original expression.

Key Concepts

Exponentiation

Exponentiation involves raising a number or variable to a power. The number or variable is called the base, and the power to which it is raised is known as the exponent. Exponents tell us how many times to multiply the base by itself. For example, in the expression \( b^8 \), the base is \( b \) and the exponent is 8, meaning \( b \) is multiplied by itself 8 times.

• Base: The number or expression that is being multiplied.
• Exponent: Denotes how many times the base is used as a factor.

In our problem, we dealt with the expression \(-\left(\frac{b^8}{625}\right)^{3/4}\), where the whole fraction \( \frac{b^8}{625} \) is raised to the power of \( \frac{3}{4} \). This requires us to apply exponents to both the numerator and the denominator, separately.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing a fraction where the numerator and the denominator are both polynomials or expressions. The first step often is to look at each part of the fraction and simplify wherever possible using exponent rules and algebraic identities.

• Numerator and Denominator Simplification: Simplify both the top and bottom parts of the fraction separately.
• Fractional Exponent: When a fraction is raised to a fractional exponent, simplify by treating the numerator and denominator individually.

In the expression \(-\frac{(b^8)^{3/4}}{(625)^{3/4}}\), notice how we simplified the fraction by computing each part separately. We applied the power to the numerator \((b^8)^{3/4} \), and the denominator \((625)^{3/4} \) was rewritten using its prime factor form to simplify further.

Power Rule

The power rule is a handy exponent rule that simplifies expressions where a power is raised to another power. The rule states that to find a power of a power, you multiply the exponents together. Mathematically, this is written as \((x^m)^n = x^{m \cdot n}\).

• Power of a Power: Multiply the exponents.
• Simplifying Computations: Aids in reducing complex expressions easily.

In our exercise, we applied the power rule when simplifying \((b^8)^{3/4}\). By multiplying 8 and \( \frac{3}{4} \), we obtained \( b^6 \), which is a much simpler expression. Similarly, for the denominator, recognizing \(625\) as \(5^4\) allowed us to apply the power rule again, simplifying it to \(5^3 = 125\). Both applications of the power rule helped us reach the final simplified form of \(-\frac{b^6}{125}\).