Question 4 of 29intermediate🔧 ApplyNumerical3 marks

Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.

Correct Answer

Step 1: Prime Factorisation 9408 = 2⁶ × 3¹ × 7²

Step 2: Check Exponents For a perfect square, all exponents must be even.

  • 2 has exponent 6 (even)
  • 3 has exponent 1 (odd)
  • 7 has exponent 2 (even)

Step 3: Find Multiplier The exponent of 3 is 1 (odd), so multiply by 3. 9408×39408 \times 3 = 28224 = 2⁶ × 3² × 7²

Step 4: Find Square Root 28224 = (2³ × 3×73 \times 7)² = 168² 28224\sqrt{28224} = 168

Answer: The smallest number to multiply 9408 by is 3. The product is 28224, and its square root is 168.

Exercise: Figure it Out (Section 1) | Q: 6 | (Page Page 10)
For More Understanding

Explanation

The chapter explains that for a number to be a perfect square, each prime factor must appear an even number of times. By prime factorisation, we identify which factors have odd exponents and multiply by those factors to make all exponents even.

Solution Steps

  1. Step 1: Find prime factorisation of 9408

  2. Step 2: 9408 = 2×47042 \times 4704 = 2² × 2352 = ... = 2⁶ × 3×73 \times 7²

  3. Step 3: Check exponents: 2 has exponent 6 (even), 3 has exponent 1 (odd), 7 has exponent 2 (even)

  4. Step 4: Multiply by 3 to make exponent of 3 even

  5. Step 5: Product = 9408×39408 \times 3 = 28224

  6. Step 6: 28224\sqrt{28224} = 26×32×72\sqrt{2⁶ × 3² × 7²} = 2³ × 3×73 \times 7 = 168

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