Problem 42 - Algebra for College Students

Question

Simplify each numerical expression. \(\left(5^{-1}-2^{-3}\right)^{-1}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{40}{3}\).

Step 1: Simplify Each Given Exponent

To start simplifying the expression \(\left(5^{-1}-2^{-3}\right)^{-1}\), we need to calculate each individual part. First, calculate \(5^{-1}\) which is the same as \(\frac{1}{5}\). Next, calculate \(2^{-3}\), which means \(\frac{1}{2^3} = \frac{1}{8}\).

Step 2: Subtract the Results

Now subtract the value of \(2^{-3}\) from \(5^{-1}\). We have \(\frac{1}{5} - \frac{1}{8}\). To subtract these fractions, find a common denominator. The least common multiple of 5 and 8 is 40. Convert \(\frac{1}{5}\) to \(\frac{8}{40}\) and \(\frac{1}{8}\) to \(\frac{5}{40}\). Subtract them: \(\frac{8}{40} - \frac{5}{40} = \frac{3}{40}\).

Step 3: Apply the Negative Exponent

Now apply the negative exponent to the result of the subtraction by inverting the fraction. Since we have \(\left(\frac{3}{40}\right)^{-1}\), this is equivalent to \(\frac{40}{3}\). This final step simplifies the expression to a simpler form.

Key Concepts

Understanding Negative Exponents

Negative exponents might seem daunting at first, but they are simpler than they appear. Unlike positive exponents, which indicate how many times a number is multiplied by itself, negative exponents suggest the reciprocal or "inverse" of the base raised to the opposite positive power. For example, if you have the expression with a negative exponent like \(5^{-1}\), it simply becomes \(\frac{1}{5}\). Similarly, \(2^{-3}\) is transformed into \(\frac{1}{2^3}\), which equals \(\frac{1}{8}\).

In general, any number \(a^{-n}\) translates into \(\frac{1}{a^n}\). This means that instead of multiplying the numbers, you turn them into fractional values. Negative exponents essentially shift numbers from being whole to being parts.

• \(a^{-1} = \frac{1}{a}\)
• \(a^{-2} = \frac{1}{a^2}\)
• Keep in mind that a negative exponent does not apply to a negative number!

Understanding this shift is crucial for anyone dealing with algebraic expressions as it lays the foundation for further simplification.

Working with Fractions

Fractions are everywhere in math, representing parts of a whole. When simplifying expressions involving negative exponents, fractions become a central element, as seen in our exercise where \(5^{-1}\) becomes \(\frac{1}{5}\) and \(2^{-3}\) becomes \(\frac{1}{8}\).

Fractions consist of a numerator, the top number, and a denominator, the bottom number. The value of a fraction tells you how many parts you're counting out of a whole divided into equal parts. When manipulating fractions, you must respect the relationship between these numerators and denominators.

• Add or subtract fractions only if they share the same denominator.
• To add or subtract, simply operate on the numerators while keeping the denominator the same.
• Multiply fractions by multiplying the numerators and the denominators respectively.

Mastering fractions paves the way for tackling more complex mathematical problems.

Finding the Least Common Denominator

When working with multiple fractions, particularly in addition or subtraction, finding the least common denominator (LCD) is essential. In our example, to subtract \(\frac{1}{5}\) and \(\frac{1}{8}\), we needed the fractions to have a common denominator. The least common multiple (LCM) of the denominators 5 and 8 is 40.

Here's how you can find the LCD:

• List multiples of each denominator to identify the smallest shared multiple.
• Convert each fraction into an equivalent fraction with this newly found denominator by multiplying both the numerator and denominator of each fraction by whatever amount needed to reach the LCD.
• Perform addition or subtraction on these adjusted fractions.

This process ensures accurate computation when fractions have different denominators.

Inverting Fractions to Simplify Expressions

Inverting fractions is a common step when dealing with expressions that include negative exponents. Inversion simply means flipping the fraction upside down. For instance, the fraction \(\left(\frac{3}{40}\right)^{-1}\) turns into \(\frac{40}{3}\).

This step usually occurs because negative exponents, by their nature, suggest inversion. Here are some important pointers:

• For a fraction \(\left(\frac{a}{b}\right)^{-1}\), the inverted form is \(\frac{b}{a}\).
• Ensure any calculations leading up to inversion have been correctly performed, as inaccurate numerators or denominators affect the final outcome.
• Inverting fractions can help conclude simplifications, giving you a final, neatly arranged result.

Paying attention to inversion can help tidy up complex algebraic expressions and lead them to their simplest form.