Question

select all the correct answers.
natalie has to list numbers that are greater than 1. which of these expressions represent numbers greater than 1?
$4^{4} \times 4^{-3}$
$8^{-2} \div 8^{-3}$
$3^{4} \div 3^{6}$
$7^{3} \times 7^{-5}$
$2^{8} \div 2^{5}$

Explanation:

Step1: Recall exponent rules

When multiplying exponents with the same base, \(a^m \times a^n = a^{m + n}\). When dividing exponents with the same base, \(a^m \div a^n = a^{m - n}\). A number \(a^b\) is greater than 1 if \(b>0\) (for \(a > 1\)), equal to 1 if \(b = 0\), and less than 1 if \(b<0\) (for \(a>1\)).

Step2: Evaluate \(4^{4}\times4^{-3}\)

Using \(a^m\times a^n=a^{m + n}\), we get \(4^{4+( - 3)}=4^{1} = 4\), which is greater than 1.

Step3: Evaluate \(8^{-2}\div8^{-3}\)

Using \(a^m\div a^n=a^{m - n}\), we get \(8^{-2-( - 3)}=8^{1}=8\), which is greater than 1.

Step4: Evaluate \(3^{4}\div3^{6}\)

Using \(a^m\div a^n=a^{m - n}\), we get \(3^{4 - 6}=3^{-2}=\frac{1}{3^{2}}=\frac{1}{9}\), which is less than 1.

Step5: Evaluate \(7^{3}\times7^{-5}\)

Using \(a^m\times a^n=a^{m + n}\), we get \(7^{3+( - 5)}=7^{-2}=\frac{1}{7^{2}}=\frac{1}{49}\), which is less than 1.

Step6: Evaluate \(2^{8}\div2^{5}\)

Using \(a^m\div a^n=a^{m - n}\), we get \(2^{8 - 5}=2^{3}=8\), which is greater than 1.

Answer:

• \(4^{4}\times4^{-3}\)
• \(8^{-2}\div8^{-3}\)
• \(2^{8}\div2^{5}\)