Problem 125 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(b^{-2}\right)^{7} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression, \((b^{-2})^7\), so it only has positive exponents. Answer: \(\frac{1}{b^{14}}\)

Step 1: Apply exponent property to simplify the expression

Use the exponent property \((a^n)^m = a^{n\cdot m}\) to multiply the exponents in the given expression: \(\left(b^{-2}\right)^{7} = b^{(-2)\cdot 7}\).

Step 2: Multiply exponents

Multiply the exponents: \((-2) \cdot 7 = -14\). This gives us the expression \(b^{-14}\).

Step 3: Use exponent property to rewrite negative exponent as positive

Apply the exponent property \(a^{-n}=\frac{1}{a^n}\) to rewrite the expression with a positive exponent: $$b^{-14} = \frac{1}{b^{14}}$$ The final expression with only positive exponents is: \(\frac{1}{b^{14}}\).

Key Concepts

Exponent Rules

Exponent rules are key to simplifying expressions in mathematics. They guide us in manipulating powers without directly computing large numbers. One crucial rule is

• Power of a Power: When raising a power to another power, multiply the exponents. For expression \[ (a^n)^m = a^{n \cdot m}, \] you take the base, keep it the same, and multiply the exponents.
• Example: For \((b^{-2})^{7}\), you multiply \(-2\) and \(7\) to get \(b^{-14}\).

If you remember these rules, simplifying expressions becomes much easier. It allows you to manage even complex equations without performing every operation step-by-step. Always keep these rules handy for quick reference.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the opposite positive power. This means that

• Rule: \(a^{-n} = \frac{1}{a^n}\)
• With a negative exponent, you're essentially 'flipping' the base.

For example, in the problem above, we have \(b^{-14}\). To make the exponent positive, you rewrite it as \(\frac{1}{b^{14}}\). The idea is that a negative exponent transforms the expression into a fraction, moving the base to the denominator.
This transformation is essential because calculations often require positive exponents for clarity and standard form. It's particularly useful in algebra and calculus, where precise expressions simplify further operations.

Simplifying Expressions

Simplifying expressions is all about finding the most straightforward form while retaining value. When dealing with exponents, this involves carefully applying rules to reduce the complexity of the expression.

Steps to simplify:

• Apply Power Rules: Start by addressing powers raised to powers, as done with the initial expression.
• Convert Negative Exponents: Change them to positive by utilizing the reciprocal.
• Ensure All Components are in the Simplest Form: After turning negative exponents positive, double-check if the expression needs further reduction.
  • Example: Converting \(b^{-14}\) to \(\frac{1}{b^{14}}\).

Simplifying isn't just about making things look neat; it's also about making calculations easier and ensuring the expression is in standard mathematical form. This process helps in solving problems more efficiently.