Problem 18 - College Algebra

Question

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. $$ 10^{6} \div\left(10^{10}\right)^{-2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(10^{26}\).

Step 1: Understand the Problem

We need to rewrite the expression \(10^{6} \div (10^{10})^{-2}\) with a single base and express it using positive exponents. This means we will manipulate the exponents and the base but won't simplify it into a single number.

Step 2: Apply the Property for Dividing Powers

The expression can be written in the form \( \frac{10^6}{(10^{10})^{-2}} \). Recall that dividing by a power is equivalent to multiplying by its reciprocal, so we change the division into multiplication of reciprocals: \(10^6 \times (10^{10})^{2}\).

Step 3: Applying the Power of a Power Rule

We have \((10^{10})^{2}\) which, by the power of a power rule, becomes \(10^{10 \times 2} = 10^{20}\). So the expression becomes \(10^{6} \times 10^{20}\).

Step 4: Combine the Powers of the Same Base

Use the property that when multiplying like bases, add the exponents: \(10^{6} \times 10^{20} = 10^{6+20}\).

Step 5: Final Expression with a Single Base

The expression is now \(10^{26}\), which is written with a single base and positive exponent as required.

Key Concepts

Power of a Power Rule

To understand the power of a power rule, consider an expression like \((a^m)^n\). This means you have a power raised to another power. The rule tells us that we can simply multiply the exponents together, becoming \(a^{m \times n}\).
In this step-by-step process, it helps to remember:

• Each exponent in the power gets multiplied by the exponent from the outer power.
• This is useful for simplifying expressions with nested exponents.
• It streamlines calculations by transforming the expression into a single power.

For example, if you have \((10^{10})^2\), the power of a power rule helps us transform it to \(10^{20}\). This reduction in complexity makes calculations and expressions easier to manage!

Dividing Powers

Dividing powers involves handling fractions like \(\frac{a^m}{a^n}\). With the same base, the rule is to subtract the exponent in the denominator from the exponent in the numerator.This gives us a simplified expression of \(a^{m-n}\).
Here's a few points to keep in mind:

• The bases must be the same for this rule to apply.
• Subtracting exponents works because division is the inverse of multiplication.
• This rule helps in breaking down complex expressions into manageable parts.

When dealing with our original exercise, rewriting \(10^6 \div (10^{10})^{-2}\)involves adjusting the division setup.By converting division into multiplication of reciprocals, you simplify to \(10^{6} \times (10^{10})^2\).This avoids needing to perform a direct division.

Multiplying Powers

When you multiply powers that have the same base, like \(a^m \times a^n\), know that you've got help from the multiplying powers rule.This handy guideline suggests adding the exponents together to streamline the expression to \(a^{m+n}\).
You'll want to remember:

• This rule only works when the bases are identical.
• Addition of exponents mirrors the multiplication of the whole numbers.
• It's a great tool for simplifying repetitive multiplicative expressions.

In the context of our exercise, the expression \(10^6 \times 10^{20}\)utilizes this rule.Naturally, it's rewritten to \(10^{26}\),demonstrating how multiple exponent combinations consolidate effortlessly.

Positive Exponents

Positive exponents reflect the straightforward application of exponent rules.When you see \(a^n\), where \(n > 0\), it simply means the base is multiplied by itself \(n\)times. This is opposed to negative exponents, which involve fractions or reciprocals.
Some key points to keep in mind:

• Positive exponents are easier to work with compared to negative ones.
• It's simpler to visualize positive exponents as repeated multiplication.
• They provide logical order and clarity when simplifying expressions.

In the given exercise, converting everything to positive exponents makes the results clear-cut.The initial expression \(10^6 \div (10^{10})^{-2}\)reframes to \(10^{26}\),illustrating the clarity when negative exponents are recast into positive terms in a single base.