Problem 150 Simplify each expression. $$\l... [FREE SOLUTION]

Chapter 6: Problem 150

Simplify each expression. $$\left(3 y^{2}\right)^{4}$$

Short Answer

Expert verified

The simplified expression is 81y^8.

Step by step solution

Understand Exponentiation

When an expression with an exponent is raised to another exponent, you multiply the exponents together. This is known as the power of a power property of exponents.

Apply the Power of a Power Property

Apply the power of a power property to the expression \ \ $(3y^2)^4$: Multiply the exponents. \ \ \[ (y^2)^4 = y^{(2*4)} = y^8 \ \ 3^1=3, so the coefficient will also be raised to the fourth power based on the same property: \ \ 3^4 = 81 \ \ Hence, we have \ (3y^2)^4 = 3^4 (y^2)^4 = 81y^8. \ \ \]

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Key Concepts

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

Exponent rules

Exponent rules are crucial for simplifying expressions efficiently. By understanding them, you can manipulate expressions to make them easier to solve. One essential rule is the power of a power property, which states that when you raise an expression to another exponent, you multiply the exponents. For example, $ (y^2)^4 = y^{2*4} = y^8 $. Another important rule is the product of powers property, which means that when you multiply two expressions with the same base, you add their exponents, like $ y^2 \times y^3 = y^{2+3} = y^5 $. These rules help you transform complex expressions into simpler forms.

Power of a power property

The power of a power property is one of the key exponentiation rules. It simplifies the process of raising an expression to another power. To apply it, you multiply the exponents together. Let's use the given exercise: $ (3y^2)^4 $. First, apply the property to the variable component: \[ (y^2)^4 = y^{2*4} = y^8 \]. Next, do the same for the coefficient: 3 has an implicit exponent of 1, so \[ 3^1 \times 4 = 3^4 \]. Calculate this as, \[ 3^4 = 81 \]. Thus, the simplified form is \[ (3y^2)^4 = 3^4(y^2)^4 = 81 y^8 \]. This method ensures consistent and error-free results.

Simplifying expressions

Simplifying expressions involves reducing them to their most basic form. This makes them easier to handle and understand. To simplify the expression $ (3y^2)^4 $, you need to correctly apply exponentiation rules. First, handle the variable part: applying the power of a power rule gives us \[ (y^2)^4 = y^8 \]. For the coefficient, \[ 3^4 = 81 \]. Put them together: \[ (3y^2)^4 = 81y^8 \]. Always ensure all components are simplified and any combined terms are accurately calculated. This ensures your final answer is as simplified as possible, providing a clear, correct solution that is easy to verify.

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91影视

Simplify each expression. $$\left(3 y^{2}\right)^{4}$$

Short Answer

Step by step solution

Understand Exponentiation

Apply the Power of a Power Property

Key Concepts

Exponent rules

Power of a power property

Simplifying expressions

One App. One Place for Learning.

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