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REET Level 1+2 . Maths

Indices / Exponents - REET Maths Notes

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Indices is one of the most calculation-heavy topics in REET Maths, where you need to apply the rules for zero power, negative power and fractional power all in the same expression. This page covers every rule, worked examples, and REET 2024's questions explained.

  • 2 questions in REET 2024
  • Class 7-8 level
  • 6 min read
Verified 17 September 2026 · Source: NCERT Class 6-8 Maths textbooks (official)

How REET asks this topic

Both Indices questions in REET 2024 were purely computational - one required adding zero power, negative power and fractions together [(2^0+2^-1+2^-2) = 7/4], the other required combining square roots, cube roots and fractional powers into a single value. In other words, this topic mainly tests whether you can apply different rules in the right order within one expression.

Laws of indices

LawFormulaExample
Multiplication lawa^m x a^n = a^(m+n)2^2 x 2^3 = 2^5 = 32
Division lawa^m / a^n = a^(m-n) (a not 0)5^4 / 5^2 = 5^2 = 25
Power of a power(a^m)^n = a^(mn)(3^2)^3 = 3^6 = 729
Zero powera^0 = 1 (a not 0)100^0 = 1
Negative powera^-n = 1/a^n2^-3 = 1/8
Power of a product(ab)^m = a^m b^m(2x3)^2 = 6^2 = 36
Power of a quotient(a/b)^m = a^m / b^m(2/3)^2 = 4/9
Fractional power (root)a^(1/n) = nth root of a8^(1/3) = 2
Combined negative and fractional powera^(-1/n) = 1/(nth root of a)8^(-1/3) = 1/2

All these rules apply directly only when the base is the same.

Understanding fractional exponents

a^(p/q) means: first take the q-th root of a, then raise it to the power p - that is, a^(p/q) = (a^(1/q))^p. REET 2024's second question used exactly this rule across three terms.

(sqrt(64))^(2/3): first sqrt(64) = 8, then 8^(2/3) = (2^3)^(2/3) = 2^2 = 4. (cube root of 64)^(3/2): first cube root of 64 = 4, then 4^(3/2) = (2^2)^(3/2) = 2^3 = 8. (5th root of 32)^0 = 1, because any non-zero number to the power 0 is always 1. Total = 4 + 8 + 1 = 13, which is the correct answer.

In the first question, 2^0 + 2^-1 + 2^-2 = 1 + 1/2 + 1/4. Making the denominator common (4) gives 4/4 + 2/4 + 1/4 = 7/4.

Common mistakes students make

Treating a^0 as 0
The correct rule: a^0 = 1 (when a is not 0), not zero.
Treating negative power as a negative number
a^-n means 1/a^n - it's a reciprocal, not a negative number. For example, 2^-2 = 1/4, not -4.
Adding exponents with different bases
a^m x b^n cannot be combined directly unless the bases (a, b) are the same; you must first make the bases equal, e.g. writing 64 as 2^6.

Standard form - a practical use of indices

Indices are used to write very large or very small numbers compactly, in what's called standard form - written as a x 10^n, where 1 <= a < 10 and n is an integer.

Example: 45,000 is written in standard form as 4.5 x 10^4, because the decimal point moves four places to the left. Similarly, 0.00032 is written as 3.2 x 10^-4, because the decimal point moves four places to the right (which is why the exponent is negative). This is also useful when teaching large numbers in science and social studies (like population or distance).

Additional worked examples

Find the value of (2^-3 x 2^5) / 2^-1. Inside the brackets, since the base is the same, add the exponents: 2^-3 x 2^5 = 2^2. Now for the division, subtract exponents: 2^2 / 2^-1 = 2^(2-(-1)) = 2^3 = 8. Solving step by step like this makes even mixed expressions manageable.

Find the value of (3^2)^-1 x 3^4. First (3^2)^-1 = 3^-2. Now 3^-2 x 3^4 = 3^(-2+4) = 3^2 = 9. It's exactly this kind of multi-step expression that REET tends to ask most, so build the habit of writing out every step separately.

Find the value of (3^-1 x 9^2) / 3^2. First convert 9^2 into base 3: 9^2 = (3^2)^2 = 3^4. Now multiply: 3^-1 x 3^4 = 3^(-1+4) = 3^3. Finally divide: 3^3 / 3^2 = 3^(3-2) = 3^1 = 3. For expressions with unequal bases, the very first step is always to convert every term to the same base.

How to teach this to children

Classroom activity

In Class 7, let students discover the rules of indices from a pattern - write 2^3=8, 2^2=4, 2^1=2 on the board and ask what comes next. Children spot the pattern (each time divide by 2) themselves and arrive at 2^0=1 and 2^-1=1/2 - so the rule is understood, not just memorised.

Teaching aids

Make exponent cards (like cards showing 2^2, 2^-1, 2^0) and have children arrange them in ascending or descending order - this visual sequencing strengthens understanding of fractional and negative powers.

Common misconception

Children often think a negative exponent means a negative answer, and that 2^0 means 0. Correct both misconceptions immediately with pattern-based worked examples.

Assessment idea

Have students solve mixed exponent expressions (like PYQ-style questions) and check whether they can apply every rule in the right order - this is the real test of multi-step understanding.

Cross-curricular link

In science class, use an example of standard form (like the speed of light, 3 x 10^8 metres/second) to show the practical importance of indices, so children don't see it as just an abstract maths rule.

Trick to remember / exam tip

In mixed exponent questions, first convert every number to the power of the same base (like 2 or 3) - this lets the multiplication/division rules apply directly, giving a faster and more accurate answer, like always treating 64 as 2^6 when solving.

Questions asked in REET 2024 (with solutions)

Question 97

The value of (2⁰ + 2⁻¹ + 2⁻²) is :

  1. A7/4
  2. B4/7
  3. C1/4
  4. D3/4

Explanation

घातांक के नियम से a⁰ = 1 और a⁻ⁿ = 1/aⁿ। अतः 2⁰ + 2⁻¹ + 2⁻² = 1 + 1/2 + 1/4 = (4 + 2 + 1)/4 = 7/4। इसलिए (A) सही है; 4/7 उल्टा (व्युत्क्रम) मान है।

Question 105

What is the value of (√64)^(2/3) + (∛64)^(3/2) + (⁵√32)⁰ ?

  1. A14
  2. B194
  3. C193
  4. D13

Explanation

(√64)^(2/3) = 8^(2/3) = (2³)^(2/3) = 2² = 4। (∛64)^(3/2) = 4^(3/2) = (2²)^(3/2) = 2³ = 8। (⁵√32)⁰ = 1, क्योंकि किसी भी शून्येतर संख्या की घात 0 होने पर मान 1 होता है। कुल मान = 4 + 8 + 1 = 13। 0 घात वाले पद को 0 मानने पर 12 आता, और घातों का गलत प्रयोग 14/193/194 जैसे उत्तर देता है।

Practice Questions

Question 1

What is the value of 5⁰?

  1. A0
  2. B1
  3. C5
  4. Dअपरिभाषित

Explanation

Any non-zero number to the power 0 is always 1, so 5^0 = 1.

Question 2

What is the value of 3⁻²?

  1. A−9
  2. B−6
  3. C1/9
  4. D9

Explanation

By the negative power rule, 3^-2 = 1/3^2 = 1/9. This is a reciprocal fraction, not a negative number.

Question 3

2³ × 2⁴ is equal to:

  1. A2⁷
  2. B2¹²
  3. C4⁷
  4. D2¹

Explanation

With the same base, exponents are added: a^m x a^n = a^(m+n), so 2^3 x 2^4 = 2^7.

Question 4

What is the value of (5²)³?

  1. A125
  2. B15625
  3. C3125
  4. D625

Explanation

By the power-of-a-power rule, (5^2)^3 = 5^6 = 15625.

Question 5

If 2ˣ = 32, what is the value of x?

  1. A4
  2. B5
  3. C6
  4. D16

Explanation

32 = 2^5, so from 2^x = 2^5, x = 5.

Question 6

What is the value of 4^(3/2)?

  1. A6
  2. B8
  3. C12
  4. D64

Explanation

4^(3/2) = (sqrt(4))^3 = 2^3 = 8.

Question 7

What is the value of (1/3)⁻³?

  1. A1/27
  2. B−27
  3. C27
  4. D9

Explanation

By the negative power rule, (1/3)^-3 = 3^3 = 27.

Question 8

aᵐ ÷ aⁿ (a ≠ 0) is equal to:

  1. Aaᵐ⁺ⁿ
  2. Baᵐ⁻ⁿ
  3. Caᵐⁿ
  4. Daⁿ⁻ᵐ

Explanation

By the division rule, exponents are subtracted with the same base: a^m / a^n = a^(m-n).

Question 9

If 3^(2x−1) = 27, what is the value of x?

  1. A1
  2. B1.5
  3. C2
  4. D2.5

Explanation

27 = 3^3, so 2x-1=3, giving 2x=4, x=2.

Question 10

Which is the most suitable teaching method to help Class 7 students naturally understand the rule a⁰=1?

  1. Aपैटर्न विधि — 2³,2²,2¹ के मान लिखकर आगे का पैटर्न स्वयं निकलवाना
  2. Bसीधे सूत्र रटवाना
  3. Cकेवल परिभाषा बोर्ड पर लिखना
  4. Dविद्यार्थियों को स्वयं समझने के लिए छोड़ देना

Explanation

In the pattern method (a constructivist approach), children see for themselves how the value changes each time you divide by the base, so the rule is understood rather than memorised - this is an effective teaching-learning method.

Frequently Asked Questions

How many questions come from Indices in REET?+

REET 2024 had 1-2 questions from this topic in the Maths section, both based on evaluating mixed exponent expressions.

What is the value of a^0?+

When a is not zero, a^0 is always 1, no matter how large or small a is.

What does a negative exponent mean?+

A negative exponent represents a reciprocal - a^-n = 1/a^n. It does not make the number negative.

Can an exponent be a fraction?+

Yes, a fractional exponent represents a root - a^(1/n) means the nth root of a.

Why is standard form used?+

It's used to write very large or very small numbers compactly and comparably, in the form a x 10^n, where 1 <= a < 10.

Can several rules of indices apply together in the same expression?+

Yes, most REET questions require applying several rules - multiplication, division, zero power, fractional power - one after another within a single expression.

Can the base of an exponent be negative?+

Yes, the base can also be negative - for example, (-2)^2 = 4 (positive, since the power is even) and (-2)^3 = -8 (negative, since the power is odd).

What is any power of 1?+

Any power of 1 (positive, negative or zero) is always 1, because multiplying 1 by itself any number of times still gives 1.

If the exponents are the same but the bases are different, how does the rule apply?+

In that case, the power-of-a-product rule applies: a^m x b^m = (ab)^m. For example, 2^3 x 3^3 = (2x3)^3 = 6^3 = 216 - here the bases (2, 3) are different but the exponent (3) is the same, so both bases are multiplied together first.

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REET 2024 Solved Paper

Read all REET 2024 questions with solutions and explanations.

Official Sources

  1. NCERT Class 6-8 Maths textbooks (official)
  2. Rajasthan Board of Secondary Education (BSER) - official REET information

Change Log

  • 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked