ICSE Class 8 Maths Numbers Chapter 07 Power and Roots

Powers and Roots

Power

A number multiplied by itself repeatedly is called a power or exponential of the number. The number is called the base of the power and the number of times it is multiplied is called the exponent or index.

The product \(a \times a\), written as \(a^2\), is a power of \(a\). The base is \(a\) and the index is 2 because \(a\) is multiplied twice. Similarly, \(x \times x \times x\), written as \(x^3\), is a power of \(x\), the base being \(x\) and the index 3 because \(x\) is multiplied thrice.

Examples

(i) In \(2^5 = 2 \times 2 \times 2 \times 2 \times 2\), base = 2, index = 5.

(ii) In \(8^3 = 8 \times 8 \times 8\), base = 8, index = 3.

Squares and Square Roots

If a number is multiplied by itself, we get a squared number, also called a perfect square. For example, \(4 \times 4 = 4^2 = 16\). So, 16 is a perfect square and 4 is said to be the square root of 16. Thus, the square root of \(a\) is \(b\) if \(b^2 = a\). The symbol for square root is \(\sqrt{\phantom{x}}\) or \(\left(\phantom{x}\right)^{1/2}\). Therefore, we write \(\sqrt{16} = 4\) or \((16)^{1/2} = 4\).

Examples

(i) \(7 \times 7 = 7^2 = 49\). So, the square root of 49 is 7. \(\therefore \sqrt{49} = 7\).

(ii) \(15 \times 15 = 15^2 = 225\). So, the square root of 225 is 15. \(\therefore \sqrt{225} = 15\).

Note \(4^2 = 16\) and \((-4)^2 = 16\). So, \(\sqrt{16} = 4\) and -4. Similarly, \(\sqrt{49} = 7\) and -7, \(\sqrt{225} = 15\) and -15, and so on. Here, we will consider only the positive value of a square root.

To Find the Square Root of a Number

There are two ways of finding the square roots of large numbers.

(a) By prime factorization - (b) By division

By Prime Factorization

Steps 1. Write the number as a product of prime factors.

2. Make pairs of equal prime factors.

3. Take one factor from each pair and multiply them.

You can also express the number as a product of powers of prime factors. Taking half of each index and then finding the product of the factors will give the square root.

Example

Find the square root of 2025.

Solution On prime factorization,

\[2025 = 3 \times 3 \times 3 \times 3 \times 5 \times 5 = (3 \times 3) \times (3 \times 3) \times (5 \times 5)\]

\[\therefore \sqrt{2025} = 3 \times 3 \times 5 = 45\]

Alternatively

\[2025 = 3^4 \times 5^2\]

\[\therefore \sqrt{2025} = 3^{4/2} \times 5^{2/2} = 3^2 \times 5 = 9 \times 5 = 45\]

By Division

Steps 1. Make pairs of the digits of the given number from right to left. If the number of digits is odd, one digit will be left unpaired at the extreme left of the number. Put a small line segment over each pair of digits.

2. Consider the first pair of digits (or the single unpaired digit) from left. This is the dividend. Find the greatest number the square of which is not more than the dividend. Write this number in the place of the quotient.

3. Write the square of the number obtained in Step 2 below the dividend and subtract. Find the remainder (if any).

4. Write the remainder obtained in Step 3 along with the next pair of digits of the given number. This new number is the new dividend. (If there is no remainder in Step 3, write only the next pair of digits of the given number.)

5. Write the first quotient just below the divisor and add them.

6. Write the largest possible digit on the right of the sum (obtained in Step 5) so that the product of the new number and the largest possible digit does not exceed the new dividend. Subtract it from the new dividend. This largest possible digit will be the second digit of the square root of the given number.

7. Write the remainder (if any) obtained in Step 6 along with the next pair of digits of the given number. This number is the new dividend. Repeat the process till all the pairs of the given number are exhausted.

Example

Find the square roots of (i) 3721 and (ii) 15376.

Solution

(i)

\(\therefore \sqrt{3721} = 61\)

Steps 1. Make the pairs of digits, that is, 21 and 37. (Here, no digit is left out of pairing.)

Teacher's Note

Understanding powers helps in calculating large quantities efficiently, such as bacterial growth in biology or compound interest in finance.

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