Question

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{6 g^{5}}{7 h^{7}}\right)^{2} $$

Short answer

Question: Simplify the expression $\left(\frac{6g^{5}}{7h^{7}}\right)^{2}$. Answer: $\frac{36g^{10}}{49h^{14}}$.

Step-by-step solution

Analyze the expression

Our goal is to remove the parentheses and simplify the expression. We have an exponent of 2 and a fraction inside the parentheses. We can apply the power rule of exponents to simplify the expression.

Apply the power rule

The power rule of exponents states that when we have a power of a fraction raised to an exponent, we can distribute that exponent to both the numerator and the denominator of the fraction. So, we have: $$ \left(\frac{6 g^{5}}{7 h^{7}}\right)^{2} = \frac{(6 g^{5})^{2}}{(7 h^{7})^{2}} $$

Apply the exponent rule

Next, we need to apply the exponent rule to each term in the numerator and denominator. The exponent rule states that when raising an expression with multiple factors to an exponent, we can distribute the exponent individually to each factor. This gives us: $$ \frac{(6 g^{5})^{2}}{(7 h^{7})^{2}} = \frac{6^{2} \cdot (g^{5})^{2}}{7^{2} \cdot (h^{7})^{2}} $$

Simplify each term

Now, we can simplify each term by calculating the numerical values and applying the power rule for the variable factors. This will give us: $$ \frac{6^{2} \cdot (g^{5})^{2}}{7^{2} \cdot (h^{7})^{2}} = \frac{36g^{10}}{49h^{14}} $$

Final Answer

After simplifying the expression and removing the parentheses, the final answer is: $$ \frac{36g^{10}}{49h^{14}} $$

Key concepts

Power Rule

The power rule is a fundamental principle in exponentiation that helps in simplifying expressions involving powers. When we have a power raised to another power, this rule states that we multiply the exponents. Consider the expression \((a^m)^n\); using the power rule, it simplifies to \(a^{m \cdot n}\).

This is especially useful when dealing with fractions inside parentheses, as we can distribute the outer exponent to both the numerator and the denominator. For example, in the expression \(\left( \frac{6g^5}{7h^7} \right)^2\), applying the power rule means we apply the 2 to each component inside the fraction:

• Numerator: \((6g^5)^2 = 6^2 \cdot (g^5)^2\)
• Denominator: \((7h^7)^2 = 7^2 \cdot (h^7)^2\)

Using the power rule simplifies the step of exponentiation, making expressions like this easier to manage and solve.

Simplification of Expressions

Simplification of expressions is about making the expression as concise as possible without changing its value. After applying the power rule, we have \(\frac{6^2 \cdot (g^5)^2}{7^2 \cdot (h^7)^2}\).

The aim is to compute any straightforward mathematical operations and simplify like terms.

• Calculate the numerical exponents: \(6^2\) becomes 36 and \(7^2\) becomes 49.
• Apply the power rule to variable factors: \((g^5)^2\) simplifies to \(g^{10}\) and \((h^7)^2\) simplifies to \(h^{14}\).

Putting it all together, we achieve the simplified expression of \(\frac{36g^{10}}{49h^{14}}\).
Breaking it down like this helps in understanding each step, showing why simplification is essential to not only solve problems efficiently but also ensure accuracy in mathematical communication.

Algebraic Fractions

Algebraic fractions, much like numerical fractions, have numerators and denominators, but include variables. Understanding their simplification is crucial when they involve exponents. An algebraic fraction can look intimidating, but by breaking them down using rules like the power rule, they become more manageable.

Key points in handling algebraic fractions include:

• Applying exponents across both numerator and denominator as needed.
• Simplifying each element independently.
• Ensuring that the expression is reduced to its simplest form without any common factors remaining.

For instance, in \(\frac{36g^{10}}{49h^{14}}\), note that there are no additional factors common to the 36 in the numerator and 49 in the denominator, nor between the variable parts of the fraction. This confirms that the expression is already simplified.
Algebraic fractions can sometimes involve more complex operations, but by systematically applying the principles of simplification, they often become just as easy to handle as simpler numeric fractions.