Revision and important questions: Power Play

6.1 Formula box

The laws below hold for non-zero bases. The scientific notation form needs a coefficient between 1 and 10.

6.2 Common mistakes

• Adding exponents when powers are multiplied by different bases. The law n^a x n^b = n^{a+b} needs the same base; 2^3 x 3^3 must be handled as (2 x 3)^3.
• Multiplying exponents in a product instead of a power of a power. Use n^a x n^b = n^{a+b}, but (n^a)^b = n^{ab}.
• Writing n^0 as 0. For every non-zero n, n^0 = 1, because n^a / n^a = 1.
• Treating a negative exponent as a negative number. 10^{-3} is a small positive fraction, 1/1000, not -1000.
• Forgetting that a negative base with an odd exponent gives a negative result; (-2)^5 = -32.
• Writing scientific notation with a coefficient that is not between 1 and 10. The form 0.59 x 10^4 is valid arithmetic but not standard; 5.9 x 10^3 is standard.
• Mixing up lakh, crore and million. A lakh is 10^5, a crore is 10^7, a million is 10^6 and a billion is 10^9.
• Rounding an estimate too early. Keep the powers of ten separate until the final step.

6.3 Important questions with answers

Question 1. Simplify 2^7 x 2^5 / 2^9.

Answer. Add the exponents in the numerator: 2^{12}. Subtract the denominator exponent: 2^3 = 8.

Question 2. Evaluate (3^{-2})^{-2}.

Answer. Multiply the exponents: 3^4 = 81.

Question 3. Write 0.00045 in scientific notation.

Answer. Move the decimal point four places: 4.5 x 10^{-4}.

Question 4. Is 2^{10} larger or smaller than 10^3?

Answer. 2^{10} = 1024, which is larger than 1000.

Question 5. A sheet 0.001 cm thick is folded 12 times. Find the thickness.

Answer. 0.001 x 2^{12} = 0.001 x 4096 = 4.096 cm.

Question 6. How many 4-digit codes can be made using digits 0 to 9 if repetition is allowed?

Answer. Each of the four positions has 10 choices, so there are 10^4 = 10,000 codes.

Question 7. Simplify m^7 / m^3 and write it without a negative exponent.

Answer. m^{7-3} = m^4.

Question 8. Write 348000000 in scientific notation.

Answer. 3.48 x 10^8.

Question 9. Which is greater, 2^{30} or 10^9? Explain.

Answer. 2^{30} = 1,073,741,824, which is slightly larger than 10^9. The power 2^{10} is close to 10^3, so 2^{30} is close to 10^9 but a little above it.

Question 10. The number of lotuses doubles daily and covers a pond on day 30. On which day is it half covered?

Answer. Day 29, because doubling once more fills the pond on day 30.

6.4 Exam strategy

Read the base and the operation before touching the exponents. If two powers of the same base are multiplied, add the exponents; if a power is raised to a power, multiply; if powers are divided, subtract. Write the law you are using in one short line, then substitute. For a numerical answer, evaluate the power once and reuse it in the next step instead of recomputing.

For scientific notation, finish by checking the coefficient. It must be at least 1 and below 10. If the coefficient is 24.6, the answer is not yet in standard form; rewrite it as 2.46 x 10^{11} by moving the point one place. A quick order check, such as comparing 2^{10} with 10^3, catches most mistakes in comparisons.

For estimation questions, state your assumptions explicitly. A reader should be able to see why 45 kg and 70 rupees per kg were chosen. The arithmetic after the assumptions is ordinary multiplication, and the answer is an estimate, not an exact measurement.

6.5 Practice set

Practice 1. Simplify 3^8 / 3^5. Answer: 3^3 = 27.

Practice 2. Simplify (2^3)^4. Answer: 2^{12} = 4096.

Practice 3. Write 0.0000072 in scientific notation. Answer: 7.2 x 10^{-6}.

Practice 4. Compare 3^{10} and 10^4. Answer: 3^{10} = 59049, which is larger than 10000.

Practice 5. A code has 4 letters, each from A to Z, with repetition allowed. How many codes are possible? Answer: 26^4 = 456976.

Practice 6. A rupee coin is 1 mm thick. How tall is a stack of 10^6 coins? Answer: 10^6 mm = 1000 m = 1 km.

6.6 Formula practice with reasoning

The laws are short, but each one has a condition. Write the condition beside the law before using it. The product law n^a x n^b = n^{a+b} needs the same base; it fails for 2^3 x 3^2, which must be evaluated as 8 x 9 = 72. The power law (n^a)^b = n^{ab} needs the outer operation to be a power, not a product; 2^3 x 2^4 uses the product law, while (2^3)^4 uses the power law.

Negative exponents follow the reciprocal rule. A good check is to write the positive power first and then take the reciprocal: 3^{-2} is the reciprocal of 9, so it is 1/9. Similarly, 10^{-5} is 1/100000. With practice, the reciprocal step becomes automatic and the negative sign no longer looks like a minus.

Scientific notation combines place value with exponents. The coefficient records the digits that matter, and the exponent records the size. When a number is written in standard form, the first digit of the coefficient is also the first non-zero digit of the original number. That gives a quick check: 70,04,00,00,000 starts with 7 and becomes 7.004 x 10^{10}, with ten zeros after the leading digit.

For growth questions, identify whether the change is additive or multiplicative before calculating. Adding 20 cm per step is linear; doubling the thickness per fold is exponential. The same distinction explains why the folded paper reaches the Moon in 46 folds while a ladder needs a huge step count.