Core Knowledge
When working with powers, we must be precise about identifying the base (what is being multiplied) and the index (how many times).
In $a^n$, $a$ is the base and $n$ is the index.
Example
Answer the following:
a)
$$\text{What is the base of } 6^5?$$
$$6$$
b)
$$\text{What is the index of } 6^5?$$
$$5$$
Question 5
On your whiteboards, answer the following:
a)
$$\text{What is the base of } 7^{2a}?$$
b)
$$\text{What is the index of } (-5)^p?$$
c)
$$\text{What is the index of } m^{n+3}?$$
d)
$$\text{What is the base of } (-b)^{4c}?$$
Answers
Show me your whiteboards.
a)
$$\text{What is the base of } 7^{2a}?$$
$$7$$
b)
$$\text{What is the index of } (-5)^p?$$
$$p$$
c)
$$\text{What is the index of } m^{n+3}?$$
$$n+3$$
d)
$$\text{What is the base of } (-b)^{4c}?$$
$$-b$$
Question 6
You haven't seen questions exactly like this before, but using what you know, on your whiteboards, answer the following:
a)
$$\text{What is the base of } -x^{2y}?$$
b)
$$\text{What is the index of } (a+b)^7?$$
c)
$$\text{What is the base of } (3p)^q?$$
d)
$$\text{What is the index of } 2^{x^2}?$$
Answers
Show me your whiteboards.
a)
$$\text{What is the base of } -x^{2y}?$$
$$x$$
b)
$$\text{What is the index of } (a+b)^7?$$
$$7$$
c)
$$\text{What is the base of } (3p)^q?$$
$$3p$$
d)
$$\text{What is the index of } 2^{x^2}?$$
$$x^2$$
Core Knowledge
We can verify statements about bases and indices by carefully identifying each part of the power.
Always check what is being multiplied (the base) and how many times (the index).
Example
Determine whether the following statements are true or false:
a)
$$\text{True or false: 3 is the base of } 3^x\text{.}$$
$$\text{True}$$
b)
$$\text{True or false: 5 is the index of } 5^y\text{.}$$
$$\text{False}$$
Question 7
On your whiteboards, write true or false for each of the following:
a)
$$6 \text{ is the base of } 6^{3n}\text{.}$$
b)
$$2p \text{ is the index of } 4^{2p}\text{.}$$
c)
$$-b \text{ is the base of } -b^c\text{.}$$
d)
$$x+1 \text{ is the index of } y^{x+1}\text{.}$$
Answers
Show me your whiteboards.
a)
$$6 \text{ is the base of } 6^{3n}\text{.}$$
$$\text{True}$$
b)
$$2p \text{ is the index of } 4^{2p}\text{.}$$
$$\text{True}$$
c)
$$-b \text{ is the base of } -b^c\text{.}$$
$$\text{False}$$
d)
$$x+1 \text{ is the index of } y^{x+1}\text{.}$$
$$\text{True}$$
Question 8
You haven't seen questions exactly like this before, but using what you know, on your whiteboards, write true or false:
a)
$$a+b \text{ is the base of } (a+b)^5\text{.}$$
b)
$$\dfrac{1}{2} \text{ is the index of } n^{\frac{1}{2}}\text{.}$$
c)
$$3 \text{ is the base of } -3^k\text{.}$$
d)
$$-x \text{ is the base of } (-x)^{y+2}\text{.}$$
Answers
Show me your whiteboards.
a)
$$a+b \text{ is the base of } (a+b)^5\text{.}$$
$$\text{True}$$
b)
$$\dfrac{1}{2} \text{ is the index of } n^{\frac{1}{2}}\text{.}$$
$$\text{True}$$
c)
$$3 \text{ is the base of } -3^k\text{.}$$
$$\text{True}$$
d)
$$-x \text{ is the base of } (-x)^{y+2}\text{.}$$
$$\text{True}$$
Question 9
On your whiteboards, answer the following:
a)
$$\text{What is the power of } 2^6?$$
b)
$$\text{What is the base of } 9^{m+1}?$$
c)
$$\text{What is the index of } (-4)^n?$$
d)
$$\text{What is the power of } x^{3p}?$$
Answers
Show me your whiteboards.
a)
$$\text{What is the power of } 2^6?$$
$$2^6$$
b)
$$\text{What is the base of } 9^{m+1}?$$
$$9$$
c)
$$\text{What is the index of } (-4)^n?$$
$$n$$
d)
$$\text{What is the power of } x^{3p}?$$
$$x^{3p}$$
Question 10
You haven't seen questions exactly like this before, but using what you know, on your whiteboards, answer the following:
a)
$$\text{What is the power of } (a+b)^2?$$
b)
$$\text{What is the base of } -5^{x^2}?$$
c)
$$\text{What is the index of } (2m)^{n-1}?$$
d)
$$\text{What is the power of } (-y)^{\frac{1}{3}}?$$
Answers
Show me your whiteboards.
a)
$$\text{What is the power of } (a+b)^2?$$
$$(a+b)^2$$
b)
$$\text{What is the base of } -5^{x^2}?$$
$$5$$
c)
$$\text{What is the index of } (2m)^{n-1}?$$
$$n-1$$
d)
$$\text{What is the power of } (-y)^{\frac{1}{3}}?$$
$$(-y)^{\frac{1}{3}}$$
Core Knowledge
To verify statements about powers, we must distinguish between the base (what is being multiplied), the index (how many times), and the power (the entire expression).
Example
Determine whether the following statements are true or false:
a)
$$\text{True or false: The power of } 6^3 \text{ is } 6^3\text{.}$$
$$\text{True}$$
b)
$$\text{True or false: The base of } 6^3 \text{ is } 3\text{.}$$
$$\text{False}$$
Question 11
On your whiteboards, write true or false for each of the following:
a)
$$\text{The power of } 4^n \text{ is } 4^n\text{.}$$
b)
$$\text{The base of } (-7)^p \text{ is } -7\text{.}$$
c)
$$\text{The index of } m^{q+2} \text{ is } m\text{.}$$
d)
$$\text{The power of } 3^{x-1} \text{ is } 3\text{.}$$
Answers
Show me your whiteboards.
a)
$$\text{The power of } 4^n \text{ is } 4^n\text{.}$$
$$\text{True}$$
b)
$$\text{The base of } (-7)^p \text{ is } -7\text{.}$$
$$\text{True}$$
c)
$$\text{The index of } m^{q+2} \text{ is } m\text{.}$$
$$\text{False}$$
d)
$$\text{The power of } 3^{x-1} \text{ is } 3\text{.}$$
$$\text{False}$$
Question 12
You haven't seen questions exactly like this before, but using what you know, on your whiteboards, write true or false:
a)
$$\text{The base of } (a+b)^{c+d} \text{ is } a+b\text{.}$$
b)
$$\text{The power of } -x^y \text{ is } -x^y\text{.}$$
c)
$$\text{The index of } (2p)^{\frac{1}{2}} \text{ is } 2p\text{.}$$
d)
$$\text{The power of } (-m)^n \text{ is } n\text{.}$$
Answers
Show me your whiteboards.
a)
$$\text{The base of } (a+b)^{c+d} \text{ is } a+b\text{.}$$
$$\text{True}$$
b)
$$\text{The power of } -x^y \text{ is } -x^y\text{.}$$
$$\text{True}$$
c)
$$\text{The index of } (2p)^{\frac{1}{2}} \text{ is } 2p\text{.}$$
$$\text{False}$$
d)
$$\text{The power of } (-m)^n \text{ is } n\text{.}$$
$$\text{False}$$
📝 Core Knowledge Review - Write in your books:
The base: The base is the number or expression being multiplied. In $5^3$, the base is $5$.
The index: The index tells us how many times the base is multiplied by itself. In $5^3$, the index is $3$.
The power: The power is the entire expression including both base and index. In $5^3$, the power is $5^3$.
Brackets matter: In $-3^k$ the base is $3$, but in $(-3)^k$ the base is $-3$. Brackets change what gets multiplied.
Independent Apply Task
Complete all questions on the worksheet showing all working out.