Problem 31 - Algebra 2

Question

Find the value of \(a b^{n}\) if \(n=3, a=2000,\) and \(b=-\frac{1}{5}\)

Step-by-Step Solution

Answer

The value is -16.

Step 1: Understand the Problem

We need to calculate the value of the expression \( a b^n \) where \( a = 2000 \), \( b = -\frac{1}{5} \), and \( n = 3 \).

Step 2: Calculate the Power of b

Using the given value of \( b = -\frac{1}{5} \) and \( n = 3 \), calculate \( b^n = \left( -\frac{1}{5} \right)^3 \). This means \( b^n = -\left( \frac{1}{5} \right)^3 = -\frac{1}{125} \).

Step 3: Multiply by a

Now, take the result from the previous step \( -\frac{1}{125} \) and multiply it by \( a = 2000 \). So, the expression becomes \(-\frac{1}{125} \times 2000 = -16 \).

Key Concepts

Negative Exponents

Negative exponents are a way to represent fractions. When you see a negative exponent, it means you should take the reciprocal of the number and then raise it to the corresponding positive power. For example, if you encounter a number like \( x^{-3} \), it's analogous to writing \( \frac{1}{x^3} \). The negative sign simply tells you to "flip" the base.
In the context of our problem, although we are not directly using a negative exponent, understanding how they work helps when dealing with fractional bases or their powers. Raising a negative base to an odd power, like in our example where \( b = -\frac{1}{5} \) and \( n = 3 \), preserves the negativity of the result, leading to a negative outcome \(-\frac{1}{125}\). This is important to remember, as the negative sign impacts the result of multiplications thereafter.

Multiplication of Fractions

Multiplying fractions might seem tricky, but it's easier than you think. You multiply straight across the tops (numerators) and the bottoms (denominators) of the fractions involved. This rule applies no matter how simple or complicated the fractions may look.
In our example, once we calculated \( b^n = \left(-\frac{1}{5}\right)^3 = -\frac{1}{125} \), multiplying \(-\frac{1}{125}\) by 2000 may look daunting. However, since 2000 can be written as \( \frac{2000}{1} \), it simplifies to \(-\frac{2000}{125}\). To make calculations simpler, find the greatest common divisor (GCD) of the numerator and the denominator to simplify the fraction. Here, \( 2000 \div 125 = 16 \), making the result \(-16\).
Understanding this process not only helps solve current problems but also strengthens your mathematical reasoning skills for future fraction-related challenges.

Integer Powers

The concept of integer powers helps simplify repeated multiplication of the same number. If you see \( b^n \), it implies you multiply the base \( b \) by itself \( n \) times.
With our problem, \( b = -\frac{1}{5} \) raised to \( n = 3 \) translates to multiplying \(-\frac{1}{5}\) by itself three times. Think of it as:

• \(-\frac{1}{5} \times -\frac{1}{5} = \frac{1}{25}\) — the negatives cancel out.
• Then, \( \frac{1}{25} \times -\frac{1}{5} = -\frac{1}{125} \) — the negative sign returns.

Mastering integer powers is crucial because it is foundational to many math operations, from simple arithmetic to more complex algebraic expressions. Recognizing how integer exponents affect both positive and negative bases will improve your mathematical fluency.