Question

consider the power 3^4. which are correct descriptions of this power? choose three correct answers. the power is the product of 3 and 4. the value of this power is 81. the power is described as 3 to the fourth power. the power means 3 factors of 4 are being multiplied. the power can be expanded as 3·3·3·3.

Explanation:

Step1: Recall the definition of power

A power \(a^{n}\) means \(a\) is multiplied by itself \(n\) times. For \(3^{4}\), the base is \(3\) and the exponent is \(4\).

Step2: Analyze each statement

• "The power is the product of 3 and 4": \(3\times4 = 12\), but \(3^{4}=3\times3\times3\times3 = 81\). So this is incorrect.
• "The value of this power is 81": \(3^{4}=3\times3\times3\times3=(3\times3)\times(3\times3)=9\times9 = 81\). This is correct.
• "The power is described as 3 to the fourth power": By the definition of exponents, \(a^{n}\) is read as " \(a\) to the \(n\)th power". Here \(a = 3\) and \(n = 4\), so \(3^{4}\) is "3 to the fourth power". This is correct.
• "The power means 3 factors of 4 are being multiplied": \(3^{4}\) means 4 factors of 3 (\(3\times3\times3\times3\)) are being multiplied, not 3 factors of 4. So this is incorrect.
• "The power can be expanded as \(3\cdot3\cdot3\cdot3\)": By the definition of \(a^{n}=\underbrace{a\times a\times\cdots\times a}_{n\text{ times}}\), for \(a = 3\) and \(n = 4\), \(3^{4}=3\cdot3\cdot3\cdot3\). This is correct.

Answer:

• The value of this power is 81.
• The power is described as 3 to the fourth power.
• The power can be expanded as \(3\cdot3\cdot3\cdot3\).