Any non-zero number raised to the power 0 equals 1. So \(2^0 = 1\). This is a fundamental law of exponents: \(a^0 = 1\) for all \(a \neq 0\).
2. Simplify: \(3^4 \times 3^2\)
A) \(3^8\)
B) \(3^6\)
C) \(9^6\)
D) \(3^2\)
Answer: B
When multiplying powers with the same base, add the exponents: \(a^m \times a^n = a^{(m+n)}\). So \(3^4 \times 3^2 = 3^{(4+2)} = 3^6\).
3. What is the value of \(5^{-2}\)?
A) \(25\)
B) \(10\)
C) \(1/25\)
D) \(-25\)
Answer: C
Negative exponent means reciprocal: \(a^{-m} = 1/a^m\). So \(5^{-2} = 1/5^2 = 1/25\). Negative exponents do NOT give negative results.
4. Simplify: \((2^3)^2\)
A) \(2^5\)
B) \(2^6\)
C) \(2^9\)
D) \(4^5\)
Answer: B
Power of a power: \((a^m)^n = a^{(m \times n)}\). So \((2^3)^2 = 2^{(3 \times 2)} = 2^6 = 64\).
5. What is \(7^1\) equal to?
A) \(1\)
B) \(0\)
C) \(7\)
D) \(49\)
Answer: C
Any number raised to the power 1 is the number itself: \(a^1 = a\). So \(7^1 = 7\). This is true for all numbers.
6. Simplify: \(10^6 / 10^2\)
A) \(10^8\)
B) \(10^4\)
C) \(10^3\)
D) \(10^{12}\)
Answer: B
When dividing powers with the same base, subtract the exponents: \(a^m / a^n = a^{(m-n)}\). So \(10^6 / 10^2 = 10^{(6-2)} = 10^4 = 10000\).
7. Express \(0.000325\) in standard form:
A) \(3.25 \times 10^{-4}\)
B) \(325 \times 10^{-6}\)
C) \(3.25 \times 10^4\)
D) \(0.325 \times 10^{-3}\)
Answer: A
Standard form (scientific notation): \(a \times 10^n\) where \(1 \le a < 10\). Move decimal 4 places right to get \(3.25\). So \(0.000325 = 3.25 \times 10^{-4}\).
8. What is the value of \((-1)^{51}\)?
A) \(1\)
B) \(51\)
C) \(-1\)
D) \(0\)
Answer: C
\( (-1) \) raised to an ODD power \( = -1 \). \( (-1) \) raised to an EVEN power \( = 1 \). Since \( 51 \) is odd, \( (-1)^{51} = -1 \).
9. Simplify: \((3/4)^2\)
A) \(6/8\)
B) \(9/16\)
C) \(3/8\)
D) \(9/8\)
Answer: B
\( (a/b)^n = a^n / b^n \). So \( (3/4)^2 = 3^2 / 4^2 = 9/16 \). Always square both numerator and denominator separately.
10. Which of the following is equal to \(8 \times 10^4 + 3 \times 10^2 + 7 \times 10^0\)?
A) \(80307\)
B) \(83007\)
C) \(80037\)
D) \(8307\)
Answer: A
\( 8 \times 10^4 = 80000 \), \( 3 \times 10^2 = 300 \), \( 7 \times 10^0 = 7 \times 1 = 7 \). Adding: \( 80000 + 300 + 7 = 80307 \). This is expanded form using powers of 10.
11. What is the value of \(2^{-3} \times 2^5\)?
A) \(2^{15}\)
B) \(2^{-2}\)
C) \(2^2\)
D) \(2^8\)
Answer: C
\( 2^{-3} \times 2^5 = 2^{(-3+5)} = 2^2 = 4 \). Use the law \( a^m \times a^n = a^{(m+n)} \) even when one exponent is negative.
12. The number \(34,50,000\) in standard form is:
A) \(3.45 \times 10^5\)
B) \(3.45 \times 10^6\)
C) \(34.5 \times 10^5\)
D) \(345 \times 10^4\)
Answer: B
\( 34,50,000 = 3,450,000 = 3.45 \times 10^6 \). Move decimal 6 places left to get \( 3.45 \) (which is between 1 and 10). So it is \( 3.45 \times 10^6 \).
15. Express \(4.738 \times 10^5\) as a usual number:
A) \(47380\)
B) \(473800\)
C) \(4738000\)
D) \(47.38\)
Answer: B
\( 4.738 \times 10^5 \) means move the decimal point 5 places to the right: \( 4.738 \rightarrow 47.38 \rightarrow 473.8 \rightarrow 4738 \rightarrow 47380 \rightarrow 473800 \). Answer is 473800.
16. Simplify: \(2^{-1} + 3^{-1}\)
A) \(1/6\)
B) \(5/6\)
C) \(6/5\)
D) \(1/5\)
Answer: B
\( 2^{-1} = 1/2 \) and \( 3^{-1} = 1/3 \). So \( 1/2 + 1/3 = 3/6 + 2/6 = 5/6 \). Always convert negative exponents to fractions first, then add.
17. Which is the correct law: \((ab)^n = \)?
A) \(a^n + b^n\)
B) \(a^n \times b\)
C) \(a^n \times b^n\)
D) \((a+b)^n\)
Answer: C
\( (ab)^n = a^n \times b^n \). The power of a product equals the product of the powers. For example: \( (2 \times 3)^2 = 6^2 = 36 = 2^2 \times 3^2 = 4 \times 9 = 36 \).
18. What is the standard form of the mass of Earth (\(6,000,000,000,000,000,000,000,000\) kg)?
A) \(6 \times 10^{24}\)
B) \(6 \times 10^{25}\)
C) \(6 \times 10^{23}\)
D) \(60 \times 10^{23}\)
Answer: A
Count the zeros: 6 followed by 24 zeros \( = 6 \times 10^{24} \) kg. Standard form requires the coefficient to be between 1 and 10, and 6 satisfies this condition.
20. The distance from Earth to Sun is \(1.496 \times 10^{11}\) m. What is this in usual form?
A) \(149,600,000\) m
B) \(14,960,000,000\) m
C) \(149,600,000,000\) m
D) \(1,496,000,000\) m
Answer: C
\( 1.496 \times 10^{11} \) means move decimal 11 places right: \( 1.496 \rightarrow 149,600,000,000 \) m. This is approximately 150 billion metres or 150 million kilometres.
Fill in the Blanks
1. \(a^m \times a^n = a^{\_\_\_\_\_\_}\)
Answer: \(m+n\)
Law of exponents: when multiplying powers with the same base, ADD the exponents. Example: \(2^3 \times 2^4 = 2^7\).
2. \(a^m / a^n = a^{\_\_\_\_\_\_}\)
Answer: \(m-n\)
Law of exponents: when dividing powers with the same base, SUBTRACT the exponents. Example: \(5^7 / 5^3 = 5^4\).
3. \((a^m)^n = a^{\_\_\_\_\_\_}\)
Answer: \(mn\)
Power of a power: multiply the exponents. Example: \((3^2)^4 = 3^8\). Never add them when there is a power of a power.
4. Any non-zero number raised to the power zero equals \_\_\_\_\_\_.
Answer: \(1\)
\(a^0 = 1\) for any non-zero \(a\). Examples: \(5^0 = 1\), \(100^0 = 1\), \((1/2)^0 = 1\). But \(0^0\) is undefined.
5. \(a^{-n} = \_\_\_\_\_\_\)
Answer: \(1/a^n\)
Negative exponent means reciprocal: \(a^{-n} = 1/a^n\). Example: \(3^{-2} = 1/3^2 = 1/9\). The result is NEVER negative.
6. The standard form of a number is written as \(a \times 10^n\) where \(a\) is between \_\_\_\_\_\_ and \_\_\_\_\_\_.
Answer: 1 and 10
In scientific notation (standard form), \(a \times 10^n\), the coefficient ‘a’ must satisfy \(1 \le a < 10\). Example: \(4.5 \times 10^3\) is standard form but \(45 \times 10^2\) is not.
7. \((-1)\) raised to an even power is always \_\_\_\_\_\_.
\( 2^{10} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 1024 \). This is an important value to memorise: \( 2^{10} \) is also called one kilobyte in computers.
9. \((a/b)^n = \_\_\_\_\_\_\)
Answer: \(a^n/b^n\)
Power of a fraction: raise both numerator and denominator to the power. Example: \( (2/3)^3 = 2^3/3^3 = 8/27 \).
10. The number \(0.00000056\) in standard form is \_\_\_\_\_\_ \( \times 10^{\_\_\_\_\_\_} \).
Answer: \(5.6 \times 10^{-7}\)
\( 0.00000056 = 5.6 \times 10^{-7} \). Move decimal 7 places to the right to get \( 5.6 \). Since we moved right, the exponent is negative \( (-7) \).
11. \(10^6\) is called one \_\_\_\_\_\_.
Answer: million
\( 10^6 = 1,000,000 = \) one million. \( 10^3 = \) thousand, \( 10^6 = \) million, \( 10^9 = \) billion, \( 10^{12} = \) trillion. These are important to know for standard form conversions.
12. If \(2^x = 32\), then \(x = \_\_\_\_\_\_.\)
Answer: \(5\)
\( 32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5 \). So \( 2^x = 2^5 \) means \( x = 5 \). Always express the number as a power of the same base to find the exponent.
13. The value of \(3^{-1} + 4^{-1} + 5^{-1}\) as a fraction is \_\_\_\_\_\_.
Answer: \(47/60\)
\( 3^{-1} = 1/3 \), \( 4^{-1} = 1/4 \), \( 5^{-1} = 1/5 \). LCM of 3,4,5 is 60. So \( 20/60 + 15/60 + 12/60 = 47/60 \).
14. Expanded form of \(52,006\) using powers of 10: \(5 \times 10^4 + 2 \times 10^3 + 0 \times 10^2 + 0 \times 10^1 + \_\_\_\_\_\_ \times 10^0\)
\( (ab)^n = a^n \times b^n \). So \( (2 \times 3)^3 = 2^3 \times 3^3 = 8 \times 27 = 216 \). Verify: \( (6)^3 = 216 \). Both give the same answer.
16. The size of a microorganism is \(0.0000005\) m. In standard form this is \_\_\_\_\_\_ \( \times 10^{-7} \) m.
Answer: \(5\)
\( 0.0000005 = 5 \times 0.0000001 = 5 \times 10^{-7} \). Move decimal 7 places to the right to get \( 5.0 \) (which is between 1 and 10). So the coefficient is 5.
\( 5^3 \times 5^{-5} = 5^{(3 + (-5))} = 5^{-2} = 1/25 \). Use law \( a^m \times a^n = a^{(m+n)} \) even with negative exponents.
18. The number 1 lakh in standard form is \_\_\_\_\_\_ \( \times 10^5 \).
Answer: \(1\)
1 lakh \( = 1,00,000 = 100,000 = 1 \times 10^5 \). The coefficient is 1 (which is between 1 and 10), and the power is 5 (5 zeros).
19. \(a^1 = \_\_\_\_\_\_\)
Answer: \(a\)
Any number raised to the power 1 equals the number itself: \( a^1 = a \). Example: \( 7^1 = 7 \), \( 100^1 = 100 \), \( (2/3)^1 = 2/3 \).
20. Express \(4.2 \times 10^{-3}\) as a usual number: \_\_\_\_\_\_
Answer: \(0.0042\)
\( 4.2 \times 10^{-3} \) means move decimal 3 places to the LEFT: \( 4.2 \rightarrow 0.42 \rightarrow 0.042 \rightarrow 0.0042 \). Negative power means the number is less than 1.
True/False Questions
1. \(2^3 \times 3^3 = 6^3\)
Answer: True
True. \( (ab)^n = a^n \times b^n \), so the reverse is also true: \( a^n \times b^n = (ab)^n \). So \( 2^3 \times 3^3 = (2 \times 3)^3 = 6^3 = 216 \).
2. \(a^0 = 0\) for any value of \(a\).
Answer: False
False. \( a^0 = 1 \) for any NON-ZERO value of \( a \). \( 0^0 \) is undefined. Example: \( 5^0 = 1 \), not 0. A very common mistake!
3. \(2^{-3}\) is a negative number.
Answer: False
False. \( 2^{-3} = 1/2^3 = 1/8 \), which is a positive fraction (not negative). Negative exponents give reciprocals, not negative numbers.
4. \((2^3)^2 = 2^5\)
Answer: False
False. \( (a^m)^n = a^{(m \times n)} \), not \( a^{(m+n)} \). So \( (2^3)^2 = 2^{(3 \times 2)} = 2^6 = 64 \), not \( 2^5 \). Multiply the exponents, do not add them.
5. \(10^0 = 1\)
Answer: True
True. Any non-zero number raised to the power 0 is 1. So \( 10^0 = 1 \). This is consistent with the law: \( 10^3 / 10^3 = 10^{(3-3)} = 10^0 = 1 \).
6. The standard form of \(56,000\) is \(5.6 \times 10^4\).
Answer: True
True. \( 56,000 = 5.6 \times 10,000 = 5.6 \times 10^4 \). The coefficient 5.6 is between 1 and 10 and the power is 4. This is correct standard form.
7. \((-2)^4 = -16\)
Answer: False
False. \( (-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 4 \times 4 = 16 \) (positive). An even power of a negative number is ALWAYS positive. \( (-2)^4 = +16 \), not \( -16 \).
8. \(2^3 \times 3^2 = 6^5\)
Answer: False
False. \( 2^3 \times 3^2 \) cannot be simplified using exponent laws because the bases are DIFFERENT. \( 2^3 \times 3^2 = 8 \times 9 = 72 \). But \( 6^5 = 7776 \). They are not equal.
9. The mass of Earth (\(6 \times 10^{24}\) kg) is greater than the mass of Moon (\(7.35 \times 10^{22}\) kg).
Answer: True
True. To compare: \( 6 \times 10^{24} \) vs \( 7.35 \times 10^{22} \). Convert to same power: \( 600 \times 10^{22} \) vs \( 7.35 \times 10^{22} \). Clearly \( 600 > 7.35 \), so Earth is heavier.
10. \(3^{-2} = -9\)
Answer: False
False. \( 3^{-2} = 1/3^2 = 1/9 \) (a positive fraction). Negative exponent gives reciprocal, not a negative number. \( -3^2 = -9 \) (different thing) but \( 3^{-2} = 1/9 \).
11. If \(5^x = 1\), then \(x = 0\).
Answer: True
True. \( 5^0 = 1 \), so \( x = 0 \). The only power that makes any non-zero base equal to 1 is the power 0.
12. \(10^3 = 1000\) and \(10^{-3} = -1000\).
Answer: False
False. \( 10^3 = 1000 \) (correct). But \( 10^{-3} = 1/10^3 = 1/1000 = 0.001 \) (a small positive number, not -1000). Negative exponents give small fractions.
13. \((a^m)^n = (a^n)^m\)
Answer: True
True. Both equal \( a^{(mn)} \). \( (a^m)^n = a^{(mn)} \) and \( (a^n)^m = a^{(nm)} = a^{(mn)} \). Multiplication is commutative so \( mn = nm \).
14. Standard form is also called scientific notation.
Answer: True
True. Standard form and scientific notation are the same thing: writing a number as \( a \times 10^n \) where \( 1 \le a < 10 \). Both terms are used in textbooks.
15. \(2^{10} > 10^3\)
Answer: True
True. \( 2^{10} = 1024 \) and \( 10^3 = 1000 \). Since \( 1024 > 1000 \), this is true. \( 2^{10} \) is slightly bigger than \( 10^3 \).
16. The exponent in \(4 \times 10^6\) is \(4\).
Answer: False
False. In \( 4 \times 10^6 \), the base is 10 and the EXPONENT is 6. The number 4 is the coefficient (or mantissa). A common confusion – the exponent belongs to the base 10.
17. \(a^m \times b^m = (ab)^m\)
Answer: True
True. This is the power of a product law in reverse: \( a^m \times b^m = (a \times b)^m = (ab)^m \). Example: \( 2^3 \times 5^3 = (2 \times 5)^3 = 10^3 = 1000 \).
18. \(0.001\) in standard form is \(1 \times 10^{-3}\).
Answer: True
True. \( 0.001 = 1/1000 = 1/10^3 = 1 \times 10^{-3} \). The coefficient is 1 (between 1 and 10) and the exponent is -3 (since we moved decimal 3 places right).
19. \(5^2 \times 5^3 = 25^5\)
Answer: False
False. \( 5^2 \times 5^3 = 5^{(2+3)} = 5^5 \), not \( 25^5 \). When multiplying same bases, add exponents and keep the SAME base. The base does not change to 25.
20. The number of seconds in a year is approximately \(3.15 \times 10^7\).
Answer: True
True. 1 year \( = 365 \) days \( \times 24 \) hours \( \times 60 \) min \( \times 60 \) sec \( = 31,536,000 \) sec \( = 3.1536 \times 10^7 \) seconds, which is approximately \( 3.15 \times 10^7 \).