Problem 29 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (y-4)^{-6} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \((y - 4)^{-6}\) using only positive exponents. Answer: \(\frac{1}{(y-4)^6}\)

Step 1: Identify the Trigonometric Problem

We examine the given expression: \((y-4)^{-6}\).

Step 2: Apply Trigonometric Identities

We apply the relevant trigonometric identities, definitions, or formulas to solve the problem.

Step 3: Simplify and Compute

We simplify the expression using known values and identities.

Step 4: State the Result

Question: Rewrite the expression \((y - 4)^{-6}\) using only positive exponents. Answer: \(\frac{1}{(y-4)^6}\)

Key Concepts

Positive Exponents

Understanding positive exponents is critical when you're dabbling with mathematical expressions, especially when converting from negative exponents. A positive exponent simply tells us how many times to multiply the base by itself. For example, 3^4 means 3 multiplied by itself 4 times, which equals 81.

When we encounter a negative exponent, we use the relationship it has with positive exponents to convert it. The rule is that a number raised to a negative exponent is equal to the reciprocal of that number raised to the corresponding positive exponent. Thus, a^{-n} = 1/a^n, which is what we used in our exercise to rewrite (y-4)^{-6} as 1/(y-4)^6. With the positive exponent, we now understand it as multiplying (y-4) by itself 6 times, all under the fraction of 1.

Exponent Rules

Exponent rules are the guiding principles that tell us how to work with numbers raised to certain powers. These are also known as the laws of exponents and are fundamental in simplifying algebraic expressions. Some of the basic rules include the product rule (a^m)(a^n) = a^{m+n}, the quotient rule a^m/a^n = a^{m-n}, and the power of a power rule (a^m)^n = a^{mn}.

The rule for negative exponents, which significantly applies to our exercise, states that a^{-n} = 1/a^n provided that a is not zero. This transformation, from negative to positive exponents, is a clean example of using exponent rules to simplify algebraic expressions. It's essential to note that these rules apply to all numbers including variables, as far as they are not equal to zero, to avoid any form of undefined expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (like addition, subtraction, multiplication, division) without an equality sign. They can include constants, coefficients, and exponents, and can be simplified or transformed using various algebra rules.

In the context of our exercise, (y-4)^{-6} is an algebraic expression with a negative exponent. By applying our exponent rules, we manipulate this expression into a more familiar form with only positive exponents. It's important in algebra to remember that like terms (those terms with the same variable raised to the same power) can only be combined. It's much easier to manipulate and simplify expressions when all the terms are expressed with positive exponents, which is the standard form that's often easier to understand and work with.