Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without using exponents. $$ \left(\frac{5^{-2} \cdot 5^{-2}}{5^{-5}}\right)^{-2} $$

Step-by-Step Solution

Answer

The simplified expression without using exponents is \( \frac{1}{25} \).

Step 1: Apply exponent rules to the numerator

In the given expression, the terms in the numerator have the same base 5 and exponents -2. According to exponent rules, when you multiply the terms with the same base, you add the exponents: \[ 5^{-2} \cdot 5^{-2} = 5^{-2 + (-2)} \] Now, simply add the exponents: \[ 5^{-4} \]

Step 2: Apply exponent rules to the whole expression

We now have the following expression: \[ \left(\frac{5^{-4}}{5^{-5}}\right)^{-2} \] According to exponent rules, when dividing terms with the same base, we subtract the exponents: \[ \frac{5^{-4}}{5^{-5}} = 5^{-4 - (-5)} \] Now, subtract the exponents: \[ 5^{1} \]

Step 3: Apply the power of a power rule

In this step, we will apply the power of a power rule. The power of a power rule states that when a power is raised to another power, we multiply the exponents: \[ (5^{1})^{-2} = 5^{1 \cdot (-2)} \] Now, multiply the exponents: \[ 5^{-2} \]

Step 4: Rewrite without an exponent

Finally, rewrite the expression without an exponent. In this case, as the exponent is negative, we can rewrite the expression as a fraction: \[ 5^{-2} = \frac{1}{5^2} \] Now, square the base in the denominator: \[ \frac{1}{25} \] So, the simplified expression without using exponents is: \[ \frac{1}{25} \]

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first, but they're easy to master once you understand the core idea. A negative exponent indicates that you are dealing with the reciprocal of the base raised to the absolute value of the exponent. For instance, if you have a term like \(5^{-2}\), it can be rewritten as \(\frac{1}{5^2}\).
This means we move the base to the denominator, making the exponent positive:

• \(5^{-1} = \frac{1}{5}\)
• \(5^{-3} = \frac{1}{5^3}\)

Whenever you encounter a negative exponent, simply think of turning it into a fraction with 1 on top! This makes calculations much easier and more intuitive.

Exponent Multiplication

When multiplying terms with exponents that share the same base, a simple rule applies: add the exponents. For example, when you see \(5^{-2} \cdot 5^{-2}\), you add the exponents -2 and -2 to get \(5^{-4}\).

This rule is based on the fact that multiplication is essentially repeated addition. Thus:

• \(a^m \cdot a^n = a^{m+n}\)

Remember this handy rule when faced with products of powers with the same base, and you'll be simplifying terms with ease.

Exponent Division

Exponent division is closely related to exponent multiplication but uses subtraction instead. If you have terms involved with division and the same base, like \(\frac{5^{-4}}{5^{-5}}\), this means subtracting the exponents:

• \(a^m / a^n = a^{m-n}\)

In this scenario, \(-4 - (-5)\) equals \(1\), hence the expression simplifies to \(5^1\). Keep in mind that subtracting a negative exponent is equivalent to adding!

Power of a Power Rule

Dealing with an exponent raised to another exponent can be simplified using the power of a power rule. This tells us to multiply the exponents together. For example, \((5^1)^{-2}\) simplifies to \(5^{1 \cdot (-2)}\), which becomes \(5^{-2}\).

For any base \(a\) and exponents \(m\) and \(n\), the rule is:

• \((a^m)^n = a^{m \cdot n}\)

Mastering this rule helps easily manage expressions needing simplification when exponents are stacked. And if you see a negative result like in \(5^{-2}\), remember that it's just \(\frac{1}{5^2}\).