Maharashtra Board Class 7 Chapter 6 Set 28 Indices Solutions

Question 1. Simplify:
(i) \( a^6 \div a^4 \)
(ii) \( m^5 \div m^8 \)
(iii) \( p^3 \div p^{13} \)
(iv) \( x^{10} \div x^{10} \)
Answer:
(i) \( a^6 \div a^4 \)
\( = a^{6-4} \)
\( = a^2 \)
(ii) \( m^5 \div m^8 \)
\( = m^{5-8} \)
\( = m^{-3} \)
(iii) \( p^3 \div p^{13} \)
\( = p^{3-13} \)
\( = p^{-10} \)
(iv) \( x^{10} \div x^{10} \)
\( = x^{10-10} \)
\( = x^0 \)
\( = 1 \)
In simple words: To simplify expressions involving division of powers with the same base, subtract the exponent of the divisor from the exponent of the dividend. If the base is divided by itself with the same exponent, the result is the base raised to the power of zero, which equals 1.

🎯 Exam Tip: Remember the rule \( a^m \div a^n = a^{m-n} \) for positive and negative exponents, and that any non-zero number raised to the power of zero is 1. Pay close attention to negative exponents in your final answer.

Question 2. Find the value of:
(i) \( (-7)^{12} \div (-7)^{12} \)
(ii) \( 7^5 \div 7^3 \)
(iii) \( \left(\frac{4}{5}\right)^3 \div \left(\frac{4}{5}\right)^2 \)
(iv) \( 4^7 \div 4^5 \)
Answer:
(i) \( (-7)^{12} \div (-7)^{12} \)
\( = (-7)^{12-12} \)
\( = (-7)^0 \)
\( = 1 \)
(ii) \( 7^5 \div 7^3 \)
\( = 7^{5-3} \)
\( = 7^2 \)
\( = 49 \)
(iii) \( \left(\frac{4}{5}\right)^3 \div \left(\frac{4}{5}\right)^2 \)
\( = \left(\frac{4}{5}\right)^{3-2} \)
\( = \left(\frac{4}{5}\right)^1 \)
\( = \frac{4}{5} \)
(iv) \( 4^7 \div 4^5 \)
\( = 4^{7-5} \)
\( = 4^2 \)
\( = 16 \)
In simple words: To find the value of these expressions, apply the rule of dividing powers with the same base by subtracting their exponents. Remember that any non-zero number or fraction raised to the power of zero is 1, and any number raised to the power of 1 is itself.

🎯 Exam Tip: Be careful with signs when dealing with negative bases. Simplify the exponent first, then evaluate the base. Ensure you perform the subtraction of exponents correctly, especially when fractions are involved as bases.