Here are the step-by-step solutions for the worksheet.
1. Calculate the value of these powers
To solve these, we multiply the number by itself as many times as the small number (exponent) says.
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Rule: If the negative number has an
odd power (like 3, 5), the answer is negative. If it has an
even power (like 2, 4), the answer is positive.
a) $(-2)^3 = -2 \times -2 \times -2 = -8$
b) $(+2)^3 = 2 \times 2 \times 2 = 8$
c) $(-9)^2 = -9 \times -9 = 81$
d) $(+9)^2 = 9 \times 9 = 81$
e) $(-4)^3 = -4 \times -4 \times -4 = -64$
f) $(+4)^3 = 4 \times 4 \times 4 = 64$
g) $(-1)^5 = -1$ (Any odd power of -1 is -1)
h) $(+1)^5 = 1$
i) $(-5)^4 = (-5 \times -5) \times (-5 \times -5) = 25 \times 25 = 625$
j) $(+5)^4 = 5 \times 5 \times 5 \times 5 = 625$
2. Work out and give the result as an only power
We use exponent rules here:
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Multiplication: Add the exponents ($x^a \cdot x^b = x^{a+b}$).
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Division: Subtract the exponents ($x^a : x^b = x^{a-b}$).
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Power of a power: Multiply the exponents ($(x^a)^b = x^{a \cdot b}$).
a) $(-5)^3 \cdot (-5)^4$
Add exponents: $3 + 4 = 7$.
Result:
$(-5)^7$
b) $(+7)^5 : (+7)^3$
Subtract exponents: $5 - 3 = 2$.
Result:
$(+7)^2$
c) $(-12)^6 \cdot (-12)^2 \cdot (-12)$
Note that $(-12)$ is the same as $(-12)^1$.
Add exponents: $6 + 2 + 1 = 9$.
Result:
$(-12)^9$
d) $[(-4)^3]^2$
Multiply exponents: $3 \times 2 = 6$.
Result:
$(-4)^6$
e) $[(+6)^3 \cdot (+6)^4] : (+6)^6$
First, do the part in brackets (add exponents): $3 + 4 = 7$. So we have $(+6)^7$.
Now divide: $7 - 6 = 1$.
Result:
$(+6)^1$ (or just $+6$)
f) $[(+3)^2]^5 : [(+3)^2 \cdot (+3)^5]$
Top part (multiply exponents): $2 \times 5 = 10$. So, $(+3)^{10}$.
Bottom part (add exponents): $2 + 5 = 7$. So, $(+3)^7$.
Now divide top by bottom (subtract exponents): $10 - 7 = 3$.
Result:
$(+3)^3$
3. Calculate square roots
We look for a number that, when multiplied by itself, equals the number inside the root. We cannot take the square root of a negative number in basic math.
a) $\sqrt{+36}$
What times itself is 36? $6 \times 6 = 36$.
Answer:
6
b) $\sqrt{-25}$
You cannot multiply a real number by itself to get a negative result.
Answer:
NO
c) $\sqrt{+121}$
What times itself is 121? $11 \times 11 = 121$.
Answer:
11
d) $\sqrt{-169}$
This is a negative number inside the root.
Answer:
NO
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Final Answer:
1. Calculate the value:
a) -8
b) 8
c) 81
d) 81
e) -64
f) 64
g) -1
h) 1
i) 625
j) 625
2. Result as a power:
a) $(-5)^7$
b) $(+7)^2$
c) $(-12)^9$
d) $(-4)^6$
e) $(+6)^1$
f) $(+3)^3$
3. Square roots:
a) 6
b) NO
c) 11
d) NO
Parent Tip: Review the logic above to help your child master the concept of power and roots worksheet.
