Problem 100 Will help you prepare for the ma... [FREE SOLUTION]

Chapter 5: Problem 100

Will help you prepare for the material covered in the next section. Simplify: \(\frac{\left(2 x^{3}\right)^{4}}{x^{10}}\)

Short Answer

Expert verified

The simplified form of the given expression \(\frac{(2 x^{3})^{4}}{x^{10}}\) is \(16 * x^2\)

Step by step solution

Apply the power of a product law

According to the power of a product law, we have \( (ab)^n = a^n * b^n \). Let's apply this law in our problem, where \a = 2, \b = x^3, and \n = 4\. Therefore, \( (2x^3)^4 = 2^4 * (x^3)^4 \)

Apply the power of a power law

According to the power of a power law, we have \( (a^n)^m = a^{n*m} \). Let's apply this law in our problem, where \a = 2, \n = 3, and \m = 4. Therefore, \( (x^3)^4 = x^{3*4} \)

Combine the results of steps 1 and 2

The expression \(\frac{(2 x^{3})^{4}}{x^{10}} = \frac{2^{4} * x^{3*4}}{x^{10}}\). Replace \(2^{4} = 16\) and \( x^{3*4} = x^{12}\) to get \(\frac{16 * x^{12}}{x^{10}}\)

Apply the law of dividing powers with the same base

According to this law, \( \frac{a^n}{a^m} = a^{n-m} \). Let's apply this law in our problem, where \a = x, \n = 12, and \m = 10. Therefore, \( \frac{16 * x^{12}}{x^{10}} = 16 * x^{12-10} \)

Final Simplification

Replace \(x^{12-10} = x^2\) to get the simplified form of the given expression, which is \(16 * x^2\)

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Key Concepts

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

Laws of Exponents

Understanding exponents is crucial in algebra. Exponents are a shorthand way to express repeated multiplication of the same number or variable. The laws of exponents help in simplifying algebraic expressions efficiently. Let's go through three primary rules:

Multiplying powers with the same base: When you multiply two numbers that have the same base, you add their powers. For instance, \( a^m \times a^n = a^{m+n} \).

Dividing powers with the same base: When you divide numbers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. This is shown as \( \frac{a^m}{a^n} = a^{m-n} \).

Power of a power law: This states that when raising a power to another power, you multiply the exponents. For example, \( (a^m)^n = a^{m\times n} \).

Understanding these basic rules allows for simplifying expressions more easily and is essential to solving complex algebra problems.

Simplification of Expressions

Simplifying algebraic expressions involves reducing them into their simplest form while keeping the value unchanged. It helps in solving algebra problems more efficiently and effectively. Here's a simple breakdown of the process:

Identify like terms: Combine terms that have identical variable parts. This makes the equation easier to manage.

Apply laws of exponents: This involves using exponent rules like the power of a product law and dividing powers law to simplify expressions involving exponents.

Perform arithmetic operations: Carry out any needed operations such as addition, subtraction, multiplication, or division to simplify the numbers.

Recheck your work: Ensure that no further simplification can be done, and verify the calculations for accuracy.

Following these steps systematically will lead to successfully simplifying any given algebraic expression while ensuring correctness.

Power of a Product Law

The power of a product law is integral to simplifying expressions where a product is raised to an exponent. This law states that when a product is raised to a power, each factor of the product is raised to that power individually. For example, if you have \((ab)^n\), this becomes \(a^n \times b^n\). Here's a closer look at how it works:

Identify the product: Recognize the factors being multiplied together inside the parentheses.

Apply the power: Distribute the exponent to each factor. For instance, \((2x^3)^4\) becomes \(2^4 \times (x^3)^4\).

Simplify further if possible: Use other exponent laws, like the power of a power law, to simplify further, as done with \((x^3)^4 = x^{12}\).

Understanding and applying this law helps in breaking down more challenging algebraic expressions into simpler components that are easier to manage and understand.

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91影视

Will help you prepare for the material covered in the next section. Simplify: \(\frac{\left(2 x^{3}\right)^{4}}{x^{10}}\)

Short Answer

Step by step solution

Apply the power of a product law

Apply the power of a power law

Combine the results of steps 1 and 2

Apply the law of dividing powers with the same base

Final Simplification

Key Concepts

Laws of Exponents

Simplification of Expressions

Power of a Product Law

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