AP Board 7th Class Maths Solutions Chapter 8 Exponents and Powers Unit Exercise

SCERT AP 7th Class Maths Solutions Pdf Chapter 8 Exponents and Powers Unit Exercise Questions and Answers.

AP State Syllabus 7th Class Maths Solutions 8th Lesson Exponents and Powers Unit Exercise

Question 1.
Answer the following.
(i) The exponential form 14⁹ should read as
Answer:
14 is raised to the power of 9.

(ii) When base is 12 and exponent is 17, it’s exponential form is _________
Answer:
12¹⁷.

(iii) The value of (14 × 21)⁰ is
Answer:
We know a⁰ = 1
So, (14 × 21)⁰ = 1

Question 2.
Express the following numbers as a product of powers of prime factors :
(i) 648
Answer:

Given 648 = 2 × 324
= 2 × 2 × 162
= 2 × 2 × 2 × 81
= 2 × 2 × 2 × 3 × 27
= 2 × 2 × 2 × 3 × 3 × 9
= 2 × 2 × 2 × 3 × 3 × 3 × 3
∴ 648 = 2³ × 3⁴

(ii) 1600
Answer:
Given 1600 = 2 × 800
= 2 × 2 × 400
= 2 × 2 × 2 × 200
= 2 × 2 × 2 × 2 × 100
= 2 × 2 × 2 × 2 × 2 × 2 × 25
= 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5

∴ 1600 = 2⁶ × 5²

(iii) 3600
Answer:

Given 3600 = 2 × 1800
= 2 × 2 × 900
= 2 × 2 × 2 × 450
= 2 × 2 × 2 × 2 × 225
= 2 × 2 × 2 × 2 × 3 × 75
= 2 × 2 × 2 × 2 × 3 × 3 × 25
= 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5
∴ 3600 = 2⁴ × 3² × 5²

Question 3.
Simplify the following using laws of exponents.
(i) a⁴ × a¹⁰
Answer:
a⁴ × a¹⁰
We know a^m × aⁿ = a^m+n
= a⁴⁺¹⁰
∴ a⁴ × a¹⁰ = a¹⁴

(ii) 18¹⁸ ÷ 18¹⁴
Answer:
18¹⁸ ÷ 18¹⁴
We Know a^m ÷ aⁿ = a^m-n
= 18^18-14
∴ 18¹⁸ ÷ 18¹⁴ = 18⁴

(iii) (x^m)⁰
Answer:
(x^m)⁰
We Know (a^m)ⁿ = a^m.n
= x^m×n = x⁰(∵ a⁰ = 1)
∴ (x^m)⁰ = 1

(iv) (6² × 6⁴) ÷ 6³
Answer:
(6² X 6⁴) ÷ 6³
We Know a^m × aⁿ = a^m+n
= (6²⁺⁴) ÷ 6³
= 6⁶ ÷ 6³
We Know a^m ÷ aⁿ = a^m-n
= 6^6-3
∴ (6² × 6⁴) ÷ 6³ = 6³

(v) \(\left(\frac{2}{3}\right)^{p}\)
Answer:
\(\left(\frac{2}{3}\right)^{p}\)
We Know \(\left(\frac{a}{b}\right)^{m}=\frac{a^{m}}{b^{m}}=\frac{2^{p}}{3^{p}}\)
∴ \(\left(\frac{2}{3}\right)^{\mathrm{p}}=\frac{2^{\mathrm{p}}}{3^{\mathrm{p}}}\)

Question 4.
Identify the greater number in each of the following andjustify your answer.
(i) 2¹⁰ or 10²
Answer:
2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
2¹⁰ = 1024
10² = 10 × 10
10² = 100
1024 > 100
So, 2¹⁰ > 10²
∴ 2¹⁰ is greater.

(ii) 5⁴ or 4⁵
Answer:
5⁴ = 5 × 5 × 5 × 5 = 625
4⁵ =4 × 4 × 4 × 4 × 4 = 1024
1024 > 625
So, 4⁵ > 5⁴
∴ 4⁵ is greater number.

Question 5.
If \(\left(\frac{4}{5}\right)^{2} \times\left(\frac{4}{5}\right)^{5}=\left(\frac{4}{5}\right)^{k}\), then find the value of ‘k’
Answer:
Given \(\left(\frac{4}{5}\right)^{2} \times\left(\frac{4}{5}\right)^{5}=\left(\frac{4}{5}\right)^{k}\)
We Know a^m × aⁿ = a^m+n
⇒ \(\left(\frac{4}{5}\right)^{2+5}=\left(\frac{4}{5}\right)^{\mathrm{k}}\)
⇒ \(\left(\frac{4}{5}\right)^{7}=\left(\frac{4}{5}\right)^{\mathrm{k}}\)
If the bases are equal powers should be equal
⇒ 7 = k
∴ k = 7

Question 6.
If 5^2p+1 ÷ 5² = 125, then find the value of ‘p’.
Answer:
Given 5^2p+1 ÷ 5² = 125
We Know a^m ÷ aⁿ = a^m-n
⇒ 5^2p+1-2 = 5 × 5 × 5
⇒ 5^2p-1 = 5³
If the bases are equal, powers should be equal.
⇒ 2p – 1 =3
⇒ 2p = 3 + 1
⇒ 2p = 4
⇒ \(\frac{2 \mathrm{p}}{2}=\frac{4}{2}\)
∴ p = 2

Question 7.
Prove that \(\left(\frac{x^{b}}{x^{c}}\right)^{a} \times\left(\frac{x^{c}}{x^{a}}\right)^{b} \times\left(\frac{x^{a}}{x^{b}}\right)^{c}\) = 1
Answer:
Given \(\left(\frac{x^{b}}{x^{c}}\right)^{a} \times\left(\frac{x^{c}}{x^{a}}\right)^{b} \times\left(\frac{x^{a}}{x^{b}}\right)^{c}\) = 1

Question 8.
Express the following numbers in the expanded form.
(i) 20068
Answer:
20068 = (2 × 10,000) + (0 × 1000) + (0 × 100) + (6 × 10) + (8 × 1)
∴ 20068 = (2 × 104) + (6 × 101) + (8 × 1)

(ii) 120718
Answer:
120718 = (1 × 1,00,000) + (2 × 10,000) + (0 × 1000) + (7 × 100) + (1 × 10) + (8 × 1)
∴ 120718 = (1 × 105) + (2 × 104) + (7 × 102) + (1 × 101) + (8 × 1)

Question 9.
Express the number appearing in the following statements in standard form :
(i) The Moon is 384467000 meters away from the Earth approximately.
Answer:
Distance of Moon from the Earth = 384467000 metres
= 3.84467000 × 100000000

Decimal is shifted eight places to the left.
= 3.84 4 67 × 10⁸ m

Distance of Moon from the earth = 3.84 4 67 × 10⁸ m

(ii) Mass of the Sun is 1 ,989,000,000,000,000,000,00000,000000 kg.
Answer:
Mass of Sun = 1 ,989,000,000,000,000,000,00000,000000 kg
= 1.989 × 1 ,000,000,000,000,000,00000,000000
= 1.989 × 10³⁰ kg
∴ Mass of Sun = 1.989 × 10³⁰ kg.

Question 10.
Lasya solved some problems of exponents and powers in the following way. Do you agree with the solution ? If not why? Justify your answer.
Answer:
(i) x³ × x² = x⁶
Answer:
No. I won’t agree with this solution.
Given, x³ × x²
We know a^m × aⁿ = a^m+n
= a³⁺²
= x⁵ which is ≠ x⁶
so, x³ × x² ≠ x⁶
We have to add to powers. But, Lasya multiplied the powers. That’s why Lasya’s solution is wrong.

(ii) (6³)¹⁰ = 6¹³
Answer:
No, (6³)¹⁰ is not equal to 6.
We know (a^m)ⁿ = a^mn
= 6^3×10
= 6³⁰ which is ≠ 6¹³
so, (6³)¹⁰ ≠ 6¹³
We have to multiply. the powers. But, Lasya added the powers. That’s why Lasya’s solution is wrong.

(iii) \(\frac{4 x^{6}}{2 x^{2}}\) = 2x³
Answer:
No, I won’t agree with this solution.
\(\frac{4 x^{6}}{2 x^{2}}=\frac{2^{2}}{2^{1}} \times \frac{x^{6}}{x^{2}}\)
We know a^m ÷ aⁿ = a^m-n
= 2^2-1 × x^6-2
= 2¹.x⁴
= 2x⁴ which is ≠ 2x³
so, \(\frac{4 x^{6}}{2 x^{2}}\) ≠ 2x³
We have to subtract the powers. But, Lasya divided the powers. That’s why Lasya’s solution is wrong.

(iv) \(\frac{3^{5}}{9^{5}}=\frac{1}{3}\)
Answer:
No. I won’t agree with this solution.
\(\frac{3^{5}}{9^{5}}=\frac{3^{5}}{\left(3^{2}\right)^{5}}\)

We Know (a^m)ⁿ = a^mn

We have to subtract the powers. But, Lasya divided the powers. That’s why Lasya’s solution is wrong.

Question 11.
Is – 2² is equal to 4? Justify your answer.
Answer:
– 2² = – (2 × 2) = – 4 ≠ 4
∴ – 2² = – 4

Question 12.
Beulah computed 2⁵ × 2¹⁰ = 2⁵⁰. Has she done it correctly? Give the reason.
Answer:
Given 2⁵ × 2¹⁰
We know a^m × aⁿ = a^m+n
= 2⁵⁺¹⁰ = 2¹⁵ ≠ 2⁵⁰
∴ 2⁵ × 2¹⁰ ≠ 2⁵⁰
Beulah did wrong.
Here we have to add the powers.
But, he multiplied the powers.

Question 13.
Rafi computed \(\frac{3^{9}}{3^{3}}\) as 3³. Has he done
Answer:
Given \(\frac{3^{9}}{3^{3}}\)
We know \(\frac{a^{m}}{a^{n}}\) = a^m-n
= 3^9-3 = 3⁶ ≠ 3³
So, \(\frac{3^{9}}{3^{3}}\) ≠ 3³
Rafi computed wrong.
Here we have to subtract the powers. But, he divided the powers.

Question 14.
Is (a²)³ equal to a⁸? Give the reason.
Answer:
Given (a²)³ equal to a⁸
We Know (a^m)ⁿ = a^mn
= (a²)³ = a^2×3 = a⁶ ≠ a⁸
∴ (a²)³ = a⁶

We have to do 2 × 3 = 6, But not
2³ = 2 × 2 × 2 = 8