$(y^2)^3$

Definition

The term $(y^2)^3$ represents the cube of the square of the variable $y$. It is a mathematical expression that involves exponents and can be evaluated using the properties of exponents, particularly the multiplication property of exponents, which is the focus of the 6.2 chapter.

5 Must Know Facts For Your Next Test

1. The expression $(y^2)^3$ can be simplified by applying the multiplication property of exponents, which states that $(a^m)^n = a^{m \times n}$.
2. In this case, the base is $y^2$, and the exponent is 3, so the simplified expression is $y^{2 \times 3} = y^6$.
3. The exponent 3 in $(y^2)^3$ indicates that the square of $y$ ($y^2$) is being multiplied by itself three times.
4. Raising a power to a power is a common operation in algebra and can be used to simplify complex expressions involving exponents.
5. Understanding the multiplication property of exponents is crucial for working with expressions like $(y^2)^3$ and other similar expressions involving powers and variables.

Review Questions

• Explain the steps to simplify the expression $(y^2)^3$.
  • To simplify the expression $(y^2)^3$, we can apply the multiplication property of exponents. The base is $y^2$, and the exponent is 3. According to the property, $(a^m)^n = a^{m \times n}$. In this case, $a = y^2$, $m = 2$, and $n = 3$. Therefore, $(y^2)^3 = y^{2 \times 3} = y^6$.
• Describe the relationship between the exponents in the expression $(y^2)^3$.
  • The expression $(y^2)^3$ involves two exponents: the exponent 2 in $y^2$, and the exponent 3 in the overall expression. The exponent 3 in $(y^2)^3$ indicates that the square of $y$ ($y^2$) is being raised to the power of 3. This means that the base $y^2$ is being multiplied by itself three times, resulting in the simplified expression $y^6$. The relationship between the exponents demonstrates the application of the multiplication property of exponents.
• Analyze how the value of the expression $(y^2)^3$ changes as the value of $y$ is varied.
  • The value of the expression $(y^2)^3$ will change as the value of $y$ is varied. Since the expression involves raising the square of $y$ to the power of 3, the value of $(y^2)^3$ will be highly sensitive to changes in the value of $y$. For example, if $y = 2$, then $(y^2)^3 = (2^2)^3 = 4^3 = 64$. If $y = 3$, then $(y^2)^3 = (3^2)^3 = 9^3 = 729$. As the value of $y$ increases, the value of the expression $(y^2)^3$ will increase exponentially due to the cube of the square of $y$.

Related terms

Exponent:

An exponent is a number that indicates how many times a base number is multiplied by itself.

Power:

A power is the result of raising a number or variable to an exponent.

Multiplication Property of Exponents: The multiplication property of exponents states that when you multiply powers with the same base, you can add the exponents.