Problem 69 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (a-4)^{3}(a-4)^{-10} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((a-4)^3 (a-4)^{-10}\) using only positive exponents. Answer: \(\frac{1}{(a-4)^7}\)

Step 1: Identify the exponent rule

We need to use the exponent rule \(a^ma^n = a^{m+n}\) to simplify the given expression.

Step 2: Simplify the given expression using the exponent rule

We have the expression \((a-4)^3 (a-4)^{-10}\). We can apply the exponent rule to combine the powers of \((a-4)\). So, $$(a-4)^3 (a-4)^{-10} = (a-4)^{3-10}$$

Step 3: Calculate the new exponent and write down the final expression

From the previous step, we have the expression \((a-4)^{3-10}\). Calculate the new exponent: $$3 - 10 = -7$$ Since we need the expression with a positive exponent, we can use the rule \(a^{-n} = \frac{1}{a^n}\) to convert it to a positive exponent: $$(a-4)^{-7} = \frac{1}{(a-4)^7}$$ So the simplified expression is: $$\frac{1}{(a-4)^7}$$

Key Concepts

Positive Exponents

Positive exponents are used to express how many times a number, known as the base, is multiplied by itself. In mathematics, when you see a number with a positive exponent, it tells you how many times to repeat that multiplication. For example, in the expression \(b^3\), the exponent is 3. This means you multiply \(b\) by itself three times: \(b \times b \times b\). The result of this multiplication is the value of \(b^3\).

• Any number to the power of 1 is itself, e.g., \( b^1 = b \).
• Any number to the power of 0 is 1, e.g., \( b^0 = 1 \), as long as \( b \) is not zero.

Positive exponents help simplify and solve mathematical expressions by making them more compact and easier to manage.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing expressions to their simplest form. This often includes combining like terms and using mathematical rules to make the expression easier to work with.
The key rule used in our original exercise is the product of powers rule: \(a^m a^n = a^{m+n}\). When you multiply terms with the same base, just add their exponents. This simplifies the process!

• For example, \( x^2 \times x^3 = x^{2+3} = x^5 \).
• Always ensure that the bases are identical before applying the rule.

By carefully applying these rules, you can turn complex expressions into simpler, more manageable forms. This process is essential for solving many algebraic problems efficiently.

Negative Exponents

Negative exponents might seem confusing at first, but they are just as easy to understand as positive ones. A negative exponent indicates a reciprocal. For any non-zero base, \( a^{-n} = \frac{1}{a^n} \). This shows that the base is on the opposite side of the fraction as it would be with a positive exponent.
For instance, \( b^{-3} = \frac{1}{b^3} \). It’s like flipping the base from the numerator to the denominator of a fraction and vice versa.

• Negative exponents are useful for expressing small numbers or dividing variables with powers more efficiently.
• Remember that flipping a fraction with negative exponents is a common technique to simplify expressions.

Understanding negative exponents allows for conversion between different forms and contributes to solutions involving fractions and division of powers.