Problem 49 - Precalculus Mathematics for Calculus

Question

Simplify the expression and eliminate any negative exponents(s). (a) \(\frac{8 a^{3} b^{-4}}{2 a^{-5} b^{5}}\) (b) \(\left(\frac{y}{5 x^{-2}}\right)^{-3}\)

Step-by-Step Solution

Answer

(a) \( \frac{4a^8}{b^9} \); (b) \( \frac{125x^6}{y^3} \)."

Step 1: Break Down the Expression (a)

The expression to simplify is \( \frac{8 a^3 b^{-4}}{2 a^{-5} b^5} \). First, divide the coefficients 8 and 2, resulting in 4.

Step 2: Simplify the Expression (a) - Variables

Using the property that when dividing like bases, subtract the exponents, for \( a \) we have \( a^{3 - (-5)} = a^{8} \), and for \( b \) we have \( b^{-4 - 5} = b^{-9} \). This results in \( 4a^8b^{-9} \).

Step 3: Eliminate Negative Exponents (a)

The simplified expression \( 4a^8b^{-9} \) still contains a negative exponent. Rewrite it as \( 4a^8 \frac{1}{b^{9}} \), giving the final result of \( \frac{4a^8}{b^9} \).

Step 4: Break Down the Expression (b)

The expression is \( \left( \frac{y}{5 x^{-2}} \right)^{-3} \). The denominator becomes \( 5x^{2} \) since \( x^{-2} \) is \( \frac{1}{x^2} \). So, we have \( \left( \frac{y}{5x^2} \right)^{-3} \).

Step 5: Apply Negative Exponent (b)

A negative exponent means take the reciprocal and raise to the positive power. Therefore, \( \left( \frac{y}{5x^2} \right)^{-3} = \left( \frac{5x^2}{y} \right)^{3} \).

Step 6: Simplify the Expression with Positive Exponent (b)

Raise each term to the power of 3: \( (5^3)(x^2)^3 \) and \( y^3 \). This gives \( 125x^6 \) in the numerator and \( y^3 \) in the denominator. So, the expression becomes \( \frac{125x^6}{y^3} \).

Key Concepts

Negative Exponents

When you come across a negative exponent, it's essentially asking you to take the reciprocal of the base and then apply the positive exponent to that. For instance, if you have a term like \( b^{-9} \), it can be rewritten as \( \frac{1}{b^9} \). This is a central aspect of simplifying expressions with negative exponents, as illustrated in the example from problem (a).

• Remember, a negative exponent flips the base to the denominator. The exponent is then applied positively.
• It's a simple way to understand it: \( x^{-n} = \frac{1}{x^n} \).

Applying this to the expression \( 4a^{8}b^{-9} \), you can eliminate the negative exponent by rewriting it, becoming \( \frac{4a^8}{b^9} \). The important takeaway here is that converting negative exponents to positive exponents often makes the expression easier to handle and read.

Fractional Exponents

Fractional exponents mean that you have both a power and a root. Fractional exponents are often seen as a straightforward alternative to using radical notation. For instance, \( x^{1/2} \) is equivalent to \( \sqrt{x} \). It's crucial to become comfortable in interchanging these forms as needed. Though the given problem does not explicitly involve fractional exponents, understanding this concept is important for more complex expressions.

• The numerator of the fractional exponent tells you the power to raise the base.
• The denominator indicates the root to take of the base.

If encountered, simplifying \( x^{m/n} \) involves calculating \( (\sqrt[n]{x})^m \) or \( (x^m)^{1/n} \). Recognizing and employing this interchange allows flexibility in manipulating expressions efficiently.

Polynomial Division

Polynomial division is similar to long division with numbers, but it's extended to algebraic polynomials. In our context, we're focusing more on simplifying expressions, such as those involving negative exponents, which simplifies to an underlying division concept.
Consider \( \frac{y}{5x^{-2}} \) in problem (b). By converting the \( x^{-2} \) into a positive exponent as \( 5x^2 \), you're initially aligning with the division of polynomial terms.

• Here, coefficients are divided normally, and like bases are divided by subtracting exponents.
• Apply this subtraction directly to easily simplify expressions, making it akin to a simple division or separation of terms.

This involves recognizing how coefficients and identical bases can be reduced similar to the division of polynomial expressions. The key is managing the exponents: subtract for division and watch for signs.

• Simplifying \( \frac{8a^{3}}{2a^{-5}} \) involves recognizing it can be reduced to \( 4a^{8} \) initially, which is akin to a simple division problem.

Grasping these principles helps streamline polynomial simplification and division.