Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \frac{3^{0} \cdot 4 x^{-2}}{16 \cdot\left(x^{2}\right)^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{4x^4}\).

Step 1: Understand the given expression

The given expression is a fraction involving powers of numbers, a variable raised to a negative exponent, and a power raised to another power: \[ \frac{3^{0} \cdot 4 x^{-2}}{16 \cdot\left(x^{2}\right)^{3}} \]

Step 2: Simplify using the properties of exponents and arithmetic

Start by applying the properties of exponents and simplifying the fractional expression step by step: - Apply the rule that any number raised to the power of 0 is equal to 1: \(3^0 = 1\) - Simplify the fraction by identifying common factors in the numerator and denominator: \(\frac{4}{16}=\frac{1}{4}\) - Apply the rule for negative exponents: \(x^{-2} = \frac{1}{x^2}\) - Apply the power of a power rule: \(\left(x^2\right)^3 = x^{2\cdot 3} = x^6\) After applying these rules, the new expression would be: \[ \frac{1\cdot\frac{1}{4} \cdot \frac{1}{x^2}}{1 \cdot x^6} \]

Step 3: Combine the terms

Now, combine the terms and simplify further: \[ \frac{\frac{1}{4} \cdot \frac{1}{x^2}}{x^6} \] Multiply the fractions in the numerator and denominator: \[ \frac{\frac{1}{4x^2}}{x^6} \]

Step 4: Apply the division rule for exponents

We want to write the expression using positive exponents only. Apply the division rule for exponents, which states that when dividing like bases, subtract the exponent in the denominator from the exponent in the numerator: \[ \frac{1}{4x^2} \cdot \frac{1}{x^6}=\frac{1}{4}x^{2-6} \]

Step 5: Simplify the resulting expression

After simplifying the exponents, we now have: \[ \frac{1}{4}x^{-4}. \] However, we need to rewrite the expression using positive exponents only. So, using the negative exponents rule: \[ \frac{1}{4}x^{-4}=\frac{1}{4} \cdot \frac{1}{x^4} \] Thus, the final simplified expression is: \[ \frac{1}{4x^4} \]

Key Concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they're quite straightforward once you understand the rule. When you see a negative exponent, it means the reciprocal of the base raised to the positive of that exponent. For example, when you have \( x^{-2} \), it translates to \( \frac{1}{x^2} \).
This is because the negative sign in the exponent indicates the inverse operation, which is why we "flip" the base to the denominator to remove the negative.

• \( x^{-a} = \frac{1}{x^a} \)
• \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)

Understanding this rule is key to being able to work with and simplify expressions involving negative exponents. It's a fundamental tool in algebra, allowing you to transform complex expressions into more manageable forms by converting them into positive exponents where needed.

Properties of Exponents

The properties of exponents are essential rules that help simplify mathematical expressions. These properties reduce the number of steps required when working with powers, making calculations quicker and easier. Here are some of the core properties you'll encounter:

• Product of Powers Property: \( a^m \cdot a^n = a^{m+n} \). This rule states that when you multiply like bases, you add their exponents.
• Power of a Power Property: \( (a^m)^n = a^{m \cdot n} \). Here, when you raise a power to another power, you multiply the exponents.
• Zero Exponent Rule: \( a^0 = 1 \), assuming \( a \) is not zero. This means that any non-zero number raised to the power of zero gives 1.

These properties make it much easier to simplify expressions, especially those like we've worked with, where terms are raised to multiple powers, or an expression involves repeated multiplication or division of similar bases.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This often includes applying the rules of exponents and arithmetic to make the expressions as straightforward as possible. For instance, when faced with complex fractions involving multiple variables and exponents, you need to carefully apply exponent rules.
Start by performing basic arithmetic operations, such as dividing like terms, to simplify the coefficients. Next, use the properties of exponents to adjust the powers of variables:

• Simplify coefficients by finding common factors and reducing them, like \( \frac{4}{16} = \frac{1}{4} \).
• Use negative exponent rules to handle inverses, converting them into positive exponents for easier manipulation.
• Combine like terms by applying the rules for multiplying and dividing exponents, simplifying further to achieve the simplest form.

By methodically applying these techniques, you can transform a complex algebraic expression into a much simpler and more comprehensible form, which is crucial for solving algebraic equations efficiently and accurately.