Problem 36 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ a^{7} b^{-8} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$a^{7}b^{-8}$$, making sure all exponents are positive. Answer: $$\frac{a^7}{b^8}$$

Step 1: Identify the term with a negative exponent

In this given expression, $$a^{7}b^{-8}$$, the term $$b^{-8}$$ has a negative exponent. Our main goal is to make the exponent positive.

Step 2: Change the negative exponent to a positive exponent by taking the reciprocal

We'll convert the negative exponent to positive by placing the term with the negative exponent in the denominator of a fraction. The expression becomes: $$\frac{a^7}{b^8}$$ Now, all exponents are positive.

Step 3: Simplify if necessary

In this case, there are no further simplifications to make, as the expression is now in its simplest form with positive exponents. So, the final answer is: $$\frac{a^7}{b^8}$$

Key Concepts

Positive Exponents

When we talk about positive exponents, we are referring to the situation where the exponent, or power, of a number expresses how many times the base is multiplied by itself. For example, in the expression \(a^7\), the number \(a\) is multiplied by itself 7 times:

• This means: \(a \times a \times a \times a \times a \times a \times a\).

Positive exponents are straightforward because they indicate repeated multiplication. When you see a positive exponent, you should be thinking of it as repeated multiplication rather than just an abstract number.
This concept helps simplify expressions and make calculations easier.

Negative Exponents

Negative exponents can initially be confusing, but they become clear with practice. A negative exponent means you are dealing with a reciprocal. That means you flip the number or variable to the other side of the fraction line. If a number with a negative exponent appears in the numerator, move it to the denominator to make the exponent positive.
For example, in the expression \(b^{-8}\), the negative exponent \(-8\) indicates a reciprocal:

• We write \(b^{-8}\) as \(\frac{1}{b^8}\).

It is important to handle negative exponents correctly, as they represent a division operation rather than multiplication. This concept is critical for simplifying expressions correctly.

Reciprocal

The reciprocal of a number is essentially its 'flip,' as seen when addressing negative exponents. In mathematical terms, the reciprocal of \(x\) is \(\frac{1}{x}\). When working with negative exponents, finding the reciprocal helps change those exponents to positive.
Here's a simple example: if you have \(b^{-8}\), its reciprocal is written as \(\frac{1}{b^8}\).
This inversion moves the term from the numerator to the denominator of a fraction, effectively changing the exponent from negative to positive. Understanding reciprocals is crucial to mastering the conversion from negative to positive exponents, ensuring proper mathematical simplification.