CBSE Class 7 Mathematics Chapter 13: Exponents and Powers - NCERT Solutions

To find the value of an expression with exponents, we multiply the base by itself the number of times indicated by the exponent.

(i) For $2^6$, the base is 2 and the exponent is 6. This means we multiply 2 by itself 6 times:

$$2^6\; =\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 2\; =\; 64$$

(ii) For $9^3$, the base is 9 and the exponent is 3. This means we multiply 9 by itself 3 times:

$$9^3\; =\; 9\; \backslash times\; 9\; \backslash times\; 9\; =\; 81\; \backslash times\; 9\; =\; 729$$

(iii) For $11^2$, the base is 11 and the exponent is 2. This means we multiply 11 by itself 2 times:

$$11^2\; =\; 11\; \backslash times\; 11\; =\; 121$$

(iv) For $5^4$, the base is 5 and the exponent is 4. This means we multiply 5 by itself 4 times:

$$5^4\; =\; 5\; \backslash times\; 5\; \backslash times\; 5\; \backslash times\; 5\; =\; 25\; \backslash times\; 25\; =\; 625$$

Answer:

(i) 64

(ii) 729

(iii) 121

(iv) 625

To express a product in exponential form, we count how many times each distinct factor appears and write it as a base raised to the power of that count.

(i) The number 6 is multiplied by itself 4 times.

$$6\; \backslash times\; 6\; \backslash times\; 6\; \backslash times\; 6\; =\; 6^4$$

(ii) The variable $t$ is multiplied by itself 2 times.

$$t\; \backslash times\; t\; =\; t^2$$

(iii) The variable $b$ is multiplied by itself 4 times.

$$b\; \backslash times\; b\; \backslash times\; b\; \backslash times\; b\; =\; b^4$$

(iv) The number 5 appears 2 times, and the number 7 appears 3 times.

$$5\; \backslash times\; 5\; \backslash times\; 7\; \backslash times\; 7\; \backslash times\; 7\; =\; 5^2\; \backslash times\; 7^3$$

(v) The number 2 appears 2 times, and the variable $a$ appears 2 times.

$$2\; \backslash times\; 2\; \backslash times\; a\; \backslash times\; a\; =\; 2^2\; \backslash times\; a^2$$

(vi) The variable $a$ appears 3 times, the variable $c$ appears 4 times, and the variable $d$ appears 1 time.

$$a\; \backslash times\; a\; \backslash times\; a\; \backslash times\; c\; \backslash times\; c\; \backslash times\; c\; \backslash times\; c\; \backslash times\; d\; =\; a^3\; \backslash times\; c^4\; \backslash times\; d^1\; =\; a^3\; c^4\; d$$

Answer:

(i) $6^4$

(ii) $t^2$

(iii) $b^4$

(iv) $5^2\; \backslash times\; 7^3$

(v) $2^2\; \backslash times\; a^2$

(vi) $a^3\; c^4\; d$

To express a number using exponential notation, we find its prime factorization. The prime factors become the bases, and the number of times each factor appears is its exponent.

(i) For 512:

We perform prime factorization:

$$512\; =\; 2\; \backslash times\; 256$$

$$256\; =\; 2\; \backslash times\; 128$$

$$128\; =\; 2\; \backslash times\; 64$$

$$64\; =\; 2\; \backslash times\; 32$$

$$32\; =\; 2\; \backslash times\; 16$$

$$16\; =\; 2\; \backslash times\; 8$$

$$8\; =\; 2\; \backslash times\; 4$$

$$4\; =\; 2\; \backslash times\; 2$$

So, $512\; =\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 2\; =\; 2^9$

(ii) For 343:

We perform prime factorization:

$$343\; =\; 7\; \backslash times\; 49$$

$$49\; =\; 7\; \backslash times\; 7$$

So, $343\; =\; 7\; \backslash times\; 7\; \backslash times\; 7\; =\; 7^3$

(iii) For 729:

We perform prime factorization:

$$729\; =\; 3\; \backslash times\; 243$$

$$243\; =\; 3\; \backslash times\; 81$$

$$81\; =\; 3\; \backslash times\; 27$$

$$27\; =\; 3\; \backslash times\; 9$$

$$9\; =\; 3\; \backslash times\; 3$$

So, $729\; =\; 3\; \backslash times\; 3\; \backslash times\; 3\; \backslash times\; 3\; \backslash times\; 3\; \backslash times\; 3\; =\; 3^6$. (Note: The source incorrectly stated $3^6$ as $36$.)

(iv) For 3125:

We perform prime factorization:

$$3125\; =\; 5\; \backslash times\; 625$$

$$625\; =\; 5\; \backslash times\; 125$$

$$125\; =\; 5\; \backslash times\; 25$$

$$25\; =\; 5\; \backslash times\; 5$$

So, $3125\; =\; 5\; \backslash times\; 5\; \backslash times\; 5\; \backslash times\; 5\; \backslash times\; 5\; =\; 5^5$

Answer:

(i) $2^9$

(ii) $7^3$

(iii) $3^6$

(iv) $5^5$