Problem 30 - Technical Mathematics with Calculus

Question

Challenge Problems. $$(5 a)^{-2}+\left(2 a^{2} b^{-4}\right)^{-2}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{1}{25a^2} + \frac{1}{4a^4b^{8}}\).

Step 1: Apply Negative Exponent

The negative exponent rule states that for any non-zero number 'n' and any integer 'k', we have \(n^{-k} = \frac{1}{n^k}\). Apply this rule to each term separately. For the first term \((5a)^{-2}\), write it as \(\frac{1}{(5a)^2}\). For the second term \((2a^2b^{-4})^{-2}\), write it as \(\frac{1}{(2a^2b^{-4})^2}\).

Step 2: Simplify Each Term Individually

Simplify the first term by squaring both the coefficient and the variable: \(\frac{1}{(5a)^2} = \frac{1}{25a^2}\). Then, simplify the second term. First square the coefficient and the variable: \((2a^2b^{-4})^2 = 4a^4b^{-8}\). Then apply the negative exponent rule to \(b^{-8}\) to get it in the denominator: \(\frac{1}{4a^4b^{8}}\).

Step 3: Write the Final Expression

Combine the simplified expressions from each term to get the final answer: \(\frac{1}{25a^2} + \frac{1}{4a^4b^{8}}\).

Key Concepts

Simplifying Expressions with Negative Exponents

When simplifying expressions that contain negative exponents, it's crucial to understand the negative exponent rule. In algebra, a negative exponent indicates that the base of the power should be reciprocated, or flipped, thereby transforming the negative exponent into a positive one. For instance, an expression like \((5a)^{-2}\) is equivalent to \(\frac{1}{(5a)^2}\). The base, in this case 5a, is non-zero, allowing us to apply the rule.

Applying this transformation step-by-step ensures clear and accurate simplification. Once you take care of the exponents, the rest of the expression often simplifies naturally. In expressions involving variables and constants, like \(a^{-2}\) or \(3^{-3}\), remember to apply the negative exponent rule to both numeric coefficients and variables alike. After changing all negative exponents to positive ones, continue simplifying by carrying out any additional arithmetic operations such as multiplication or division as required.

Applying Exponent Rules

To confidently tackle algebraic problems with exponents, it's essential to master several exponent rules. Apart from the negative exponent rule, there are rules for multiplying and dividing powers with the same base, raising a power to another power, and distributing a power across products or quotients.

One critical rule to remember is that when you raise a power by another power, as seen in an expression like \((2a^2b^{-4})^{-2}\), you multiply the exponents. In this case, we multiply the exponents of each factor inside the parentheses by -2. It's important to comprehensively apply these rules - ensuring that every exponent is addressed. This may result in further negative exponents, requiring us to invoke the negative exponent rule again to simplify fully.

Solving Exponent Problems in Algebra

Exponent problems can appear daunting, but with a systematic approach, they are quite manageable. Start by applying the basic exponent rules correctly. This includes not only transforming negative exponents to positive by flipping the base but also paying attention to the order of operations - also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Once all exponents are positive, simplify the expression term-by-term. Combine like terms if possible, and if you arrive at fractions, look for common denominators to combine them. Solving exponent problems requires careful attention to detail so as not to miss any negative signs or exponents, which could lead to incorrect simplification. Practice is key; the more problems you solve, the more skilled you'll become at navigating the rules and operations needed to simplify and solve exponent problems in algebra.