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Chapter 10: Problem 640

Simplify each expression. $$\left(\frac{x^{3} \cdot x^{9}}{x^{5}}\right)^{2}$$

Short Answer

Expert verified
The simplified expression is \(x^{14}\).

Step by step solution

01

Simplify Inside the Parentheses

First, simplify the expression inside the parentheses. Combine the exponents in the numerator: \ \frac{x^{3} \times x^{9}}{x^{5}} \ Combine the exponents in the numerator using the property \(a^m \times a^n = a^{m+n}\): \ \frac{x^{3+9}}{x^{5}} \ This simplifies to: \ \frac{x^{12}}{x^{5}}
02

Simplify the Fraction

Now, simplify the fraction by subtracting the exponent in the denominator from the exponent in the numerator using the property \(\frac{a^m}{a^n} = a^{m-n}\): \ x^{12-5} \ Which simplifies to: \ x^{7}
03

Apply the External Exponent

Finally, apply the external exponent by raising the simplified expression to the power of 2. Use the property \((a^m)^n = a^{m \times n}\): \ (x^7)^2 \ This gives us: \ x^{7 \times 2} \ Simplifying it further: \ x^{14}

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Exponents
Understanding the properties of exponents is crucial when simplifying expressions. Here are some key properties that you should know:

First, when multiplying two expressions with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
Next, when dividing two expressions with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
Finally, when raising an expression to a power, multiply the exponents: \((a^m)^n = a^{m \times n}\).

Using these properties helps simplify complex algebraic expressions, saving time and reducing the chances of errors.
Combining Exponents
When simplifying algebraic expressions, it鈥檚 common to combine exponents.
For example, in the given exercise, \(x^{3} \times x^{9}\) combines to give \(x^{3+9} = x^{12}\).
This is possible because of the multiplication property of exponents we discussed earlier.

Similarly, when we divide \(\frac{x^{12}}{x^{5}}\), we subtract the exponent in the denominator from the exponent in the numerator: \(x^{12-5} = x^{7}\). This step ensures the expression becomes simpler before applying any external exponents.
Always remember: when combining exponents, follow the fundamental rules of adding, subtracting, or multiplying the exponents appropriately.
Simplifying Fractions in Algebra
Simplifying fractions is another essential skill in algebra. To simplify fractions involving exponents, you'll primarily use the exponent subtraction rule.
In the exercise, \(\frac{x^{12}}{x^{5}}\) becomes \(x^{12-5} = x^{7}\).
After that, we raised the resulting expression \(x^{7}\) to the power of 2, which involves another key exponent property.

You multiply the exponents: \((x^7)^2 = x^{7 \times 2} = x^{14}\).
This final result is the simplest form of the original expression, illustrating how to effectively reduce and manage algebraic fractions step-by-step.
Understanding and correctly applying these rules is essential for mastering algebra.

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