Question

Write with a single exponent. $$ \frac{(g+h)^{6}(g+h)^{5}}{\left((g+h)^{2}\right)^{4}} $$

Short answer

Question: Simplify the expression \(\frac{(g+h)^{6}(g+h)^{5}}{\left((g+h)^{2}\right)^{4}}\). Answer: The simplified expression is \((g+h)^{3}\).

Step-by-step solution

Simplify the numerator

To simplify the numerator, we need to add the exponents of the (g+h) terms, since they have the same base. Remember the rule: \((a^m)(a^n) = a^{m+n}\). $$ (g+h)^{6}(g+h)^{5} = (g+h)^{6+5} $$

Simplify the denominator

To simplify the denominator, we need to apply the power rule: \((a^m)^n = a^{mn}\). In this case, \((g+h)^{2}\) raised to the power of 4: $$ \left((g+h)^{2}\right)^{4} = (g+h)^{2\cdot4} $$

Divide the numerator by the denominator

Now that we have simplified both the numerator and the denominator, we'll divide the numerator by the denominator. Recall the rule for division of exponents with the same base: \(\frac{a^m}{a^n} = a^{m-n}\). In our case, the base is (g+h), so we have: $$ \frac{(g+h)^{11}}{(g+h)^{8}} = (g+h)^{11-8} $$

Write the final result

Finally, we write the simplified expression with a single exponent: $$ (g+h)^{3} $$

Key concepts

Exponent rules

Exponents, also known as powers, are a way of representing repeated multiplication of a number by itself. They are handy tools in mathematics that help us simplify expressions and perform calculations more efficiently. When dealing with exponents, there are several key rules you should keep in mind to simplify expressions confidently.

First, the **Multiplication Rule** is used when you have the same base being multiplied with different exponents. The base stays the same, and you add up the exponents. In mathematical terms, this rule is expressed as:

• \((a^m)(a^n) = a^{m+n}\)

Using this rule can turn a complicated expression into something much simpler.

Next, there's the **Division Rule**, which applies when you are dividing two expressions with the same base. Instead of adding the exponents, you subtract the exponent of the denominator from the exponent of the numerator:

• \(\frac{a^m}{a^n} = a^{m-n}\)

This rule is particularly helpful for simplifying fractions with exponents.

Finally, the **Power Rule** is essential when an exponent is raised to another power. This means multiplying the two exponents together:

• \((a^m)^n = a^{mn}\)

Understanding these exponent rules transforms complex expressions into simple forms through a series of logical steps.

Simplifying expressions

Simplifying expressions with exponents involves employing the rules we have discussed to consolidate the expression into an easier-to-read form. The goal of simplification is to reduce complexity and make subsequent calculations or analysis more straightforward.

Consider our original expression. To simplify it, we applied the multiplication and division rules:

First, we simplified the numerator by applying the multiplication rule to combine the exponents:

• \((g+h)^{6+5} = (g+h)^{11}\)

This action turned two separate terms into one single term with a combined exponent.

Next, the denominator was simplified using the power rule. By taking a power of a power, we found:

• \((g+h)^{2\cdot4} = (g+h)^8\)

By breaking down the calculation into smaller steps, we simplified components separately, making our work clearer moving forward.

Finally, we applied the division rule to the entire fraction and subtracted the exponents:

• \(\frac{(g+h)^{11}}{(g+h)^{8}} = (g+h)^{3}\)

Each step of simplification helped reduce the expression, allowing us to represent multiple operations within a single exponent.

Power rule

The power rule appears often in expressions involving exponents. It is especially useful in situations where an exponent itself is being raised to another power, enabling significant simplification with just one step.

The rule is written as:

• \((a^m)^n = a^{mn}\)

What this means is that you take the base's exponent and multiply it by the power to which it is raised.

In our exercise, the expression \((g+h)^{2}\) was raised to the fourth power. By applying the power rule, we calculated:

• \((g+h)^{2\cdot4} = (g+h)^8\)

This transformation from repeated multiplication to an expression with a single exponent greatly simplifies the problem.

Using the power rule not only makes handling expressions more manageable but also expedites calculations and helps in deriving solutions without drowning in excessive detail. This principle is a key to efficiently working with exponents no matter the complexity of the context.