Question

powers and roots

1. what is the value of 7^7?
2. what is an equivalent expression to 5^4 x 5^7?
3. simplify

a) 10^50 + 10^5
c) (x^3 · x^6)^2
b) \frac{(-2)^8}{-2}
d) (4n^4)^3

1. simplify

a) \sqrt{49}
b) \sqrt{144}

1. circle which of the following is not a perfect square.

a) 196
b) 256
c) 476
d) 676

1. which of the following numbers has a square root that falls between 8.2 and 9.9?

a) 9
b) 64
c) 81
d) 100

1. evaluate (-\frac{2}{5})^3

factors and multiples

1. what are all of the factors of 24?
2. list all of the prime factors of 120. (you might use a factor tree here)

Explanation:

Step1: Solve 7^7

Use the exponent - operation.
$7^7=7\times7\times7\times7\times7\times7\times7 = 823543$

Step2: Find equivalent of 5^4×5^7

Apply the rule $a^m\times a^n=a^{m + n}$.
$5^4\times5^7=5^{4 + 7}=5^{11}$

Step3: Simplify 10^{20}\div10^5

Apply the rule $a^m\div a^n=a^{m - n}$.
$10^{20}\div10^5=10^{20 - 5}=10^{15}$

Step4: Simplify $\frac{(-2)^8}{-2}$

Apply the rule $a^m\div a^n=a^{m - n}$.
$\frac{(-2)^8}{-2}=(-2)^{8-1}=(-2)^7=-128$

Step5: Simplify $(x^3\cdot x^6)^2$

First, use $a^m\cdot a^n=a^{m + n}$ inside the parentheses: $x^3\cdot x^6=x^{3 + 6}=x^9$. Then, use $(a^m)^n=a^{mn}$: $(x^9)^2=x^{9\times2}=x^{18}$

Step6: Simplify $(4n^4)^3$

Use $(ab)^n=a^n\cdot b^n$ and $(a^m)^n=a^{mn}$.
$(4n^4)^3=4^3\cdot(n^4)^3=64n^{4\times3}=64n^{12}$

Step7: Simplify $\sqrt{49}$

Find the square - root.
$\sqrt{49} = 7$

Step8: Simplify $\sqrt{144}$

Find the square - root.
$\sqrt{144}=12$

Step9: Determine non - perfect square

Check the square - roots: $\sqrt{196}=14$, $\sqrt{256}=16$, $\sqrt{476}\approx21.82$, $\sqrt{676}=26$. So 476 is not a perfect square.

Step10: Find number with square - root between 8.2 and 9.9

$\sqrt{9}=3$, $\sqrt{64}=8$, $\sqrt{81}=9$, $\sqrt{100}=10$. The number 81 has a square - root 9 which is between 8.2 and 9.9.

Step11: Evaluate $(-\frac{2}{5})^3$

Use $(a/b)^n=a^n/b^n$.
$(-\frac{2}{5})^3=\frac{(-2)^3}{5^3}=\frac{-8}{125}=-\frac{8}{125}$

Step12: Find factors of 24

$24=1\times24=2\times12=3\times8=4\times6$. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

Step13: Find prime factors of 120

Use a factor - tree: $120 = 2\times60=2\times2\times30=2\times2\times2\times15=2\times2\times2\times3\times5$. The prime factors of 120 are 2, 3, 5.

Answer:

1. 823543
2. $5^{11}$
3. a) $10^{15}$; b) - 128; c) $x^{18}$; d) $64n^{12}$
4. a) 7; b) 12
5. c) 476
6. c) 81
7. $-\frac{8}{125}$

Factors and Multiples:

1. 1, 2, 3, 4, 6, 8, 12, 24
2. 2, 3, 5