Problem 114 - College Algebra

Question

Simplify each exponential expression.Assume that variables represent nonzero real numbers. $$\frac{\left(2^{-1} x^{-3} y^{-1}\right)^{-2}\left(2 x^{-6} y^{4}\right)^{-2}\left(9 x^{3} y^{-3}\right)^{0}}{\left(2 x^{-4} y^{-6}\right)^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( 64 x^{10} / y^{8} \).

Step 1: Utilize Exponent Rules

The first step is to utilize the rules of exponents to simplify each term wherever possible. Applying power of a power rule where each factor in a base is raised individually to the outside power and applying negative base rule where the base switches its position from numerator to denominator or vice versa, we get \( 2^{2} x^{6} y^{2} * 2^{2} x^{12} y^{-8} * 1 / 2^{2} x^{-8} y^{12} \).

Step 2: Simplifying Exponents

Now, multiply the terms with same base and subsequently add their exponents, and divide subsequently subtract their exponents when divided, we get \( 2^{6} x^{10} y^{-8} \).

Step 3: Resolve Negative Exponents

Notice that y has a negative exponent. The rule is that anything with a negative exponent moves to the denominator. Therefore, our expression ends up as \( 2^{6} x^{10} / y^{8} \).

Step 4: Expanding Powers

As the last step, expand the powers of 2 and we get \( 64 x^{10} / y^{8} \).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial for simplifying exponential expressions efficiently. These rules are sets of guidelines that help us manipulate expressions involving powers in a systematic way. For instance, when you multiply exponents with the same base, you add their powers. Conversely, when you divide them, you subtract the powers. An important rule to remember is that anything raised to the power of zero is equal to one.

Using these rules not only simplifies complex expressions but also ensures accuracy in your calculations. It's like having a toolkit that makes handling numbers more structured and less intimidating. By mastering exponent rules, you can tackle various mathematical problems with confidence.

Negative Exponents

Dealing with negative exponents can initially seem confusing, but it's actually a straightforward concept once you understand the core principle. Negative exponents indicate that the base should be reciprocated (flipped) and then raised to the absolute value of the exponent. In essence, a negative exponent tells you to take the reciprocal of the base and make the exponent positive. For example, \( x^{-3} \) is the same as \( 1/x^{3} \).

Remember, negative exponents do not make the number negative; they simply change the position of the number from the numerator to the denominator or vice versa. This adjustment helps in keeping our expressions tidy and easily manageable, showcasing the efficiency of mathematical notations in capturing complex ideas with elegance and precision.

Power of a Power Rule

The power of a power rule is a fascinating concept in the realm of exponents. It occurs when you have an exponent raised to another exponent, such as \( (x^m)^n \). In these cases, you multiply the exponents together. So, \( (x^m)^n \) becomes \( x^{mn} \).

This rule beautifully demonstrates how exponents are shorthand for repeated multiplication and how they can be manipulated to simplify expressions that might otherwise look daunting. Simplifying expressions using the power of a power rule reduces them to a more manageable form, making it easier to understand and solve mathematical problems involving exponents.