Exponents and Standard Form
Exponents and Standard Form
Exponents close the number system chapter and supply 2 questions, often combined with standard form or with a comparison of large numbers. The laws are few and fixed, and almost every mistake in this topic comes from one of two specific confusions.
The Laws of Exponents
| Law | Rule |
|---|---|
| Product of same base | aⁿ × aᵐ = aⁿ⁺ᵐ |
| Quotient of same base | aⁿ ÷ aᵐ = aⁿ⁻ᵐ |
| Power of a power | (aⁿ)ᵐ = aⁿᵐ |
| Power of a product | (ab)ⁿ = aⁿ bⁿ |
| Zero exponent | a⁰ = 1, for a ≠ 0 |
| Negative exponent | a⁻ⁿ = 1 ÷ aⁿ |
The first two apply only when the bases are the same. You cannot combine 2³ and 3² by adding or multiplying exponents, and options are written specifically to tempt that move.
✗ (a³)² = a⁵ | ✓ (a³)² = a⁶ — powers of a power multiply, they do not add
Zero and Negative Exponents
Any non-zero number to the power zero is 1. The reason is worth knowing because it is often asked as reasoning: aⁿ ÷ aⁿ equals 1, and by the quotient law it also equals a⁰.
A negative exponent means a reciprocal, not a negative number. So 2⁻³ = 1/8, which is positive. Confusing a negative exponent with a negative value is the most common error in this topic.
✗ 2⁻³ = −8 | ✓ 2⁻³ = 1/2³ = 1/8, a positive number
Standard Form
Standard form: k × 10ⁿ, where 1 ≤ k < 10
Standard form, also called scientific notation, expresses very large and very small numbers compactly.
- 59,000 becomes 5.9 × 10⁴.
- 0.00032 becomes 3.2 × 10⁻⁴.
- Moving the point left gives a positive power; moving it right gives a negative power.
- The value of k must be at least 1 and less than 10 — 59 × 10³ is not standard form.
The last condition is what a question actually tests: several options will have the right value but only one will be correctly standardised.
Comparing Large Numbers
To compare numbers in standard form, compare the powers of ten first; only if they are equal do you compare the values of k. This is quicker than converting either number back, and it is the skill the question is really checking.
Exponents also connect forward: they are the language of squares and cubes from the previous topic, and of the algebraic identities in the next chapter.
ऊणात्मक घातांक का अर्थ व्युत्क्रम है, ऊणात्मक संख्या नहीं।
TRE pointer: Two traps carry this topic. The first is treating a negative exponent as a negative number; the second is adding exponents in a power of a power instead of multiplying. Both appear as ready-made wrong options. In a teaching paper you may also be asked why a⁰ equals 1, which needs the quotient-law argument rather than a memorised line. With five options and −1/3 for a wrong answer, reasoning items like this are worth the extra fifteen seconds.
60-Second Recap
- Same base: multiply means add exponents, divide means subtract.
- Power of a power multiplies the exponents.
- a⁰ = 1 for any non-zero a, provable from the quotient law.
- A negative exponent means a reciprocal, and the value stays positive.
- Standard form is k × 10ⁿ with k at least 1 and less than 10.
- Compare powers of ten first when comparing numbers in standard form.
Frequently Asked Questions
Why is any number to the power zero equal to one?
Because a to the power n divided by a to the power n is clearly one, and by the quotient law the same expression equals a to the power zero. So a to the power zero must be one, for any non-zero a. Zero to the power zero is left undefined.
Does a negative exponent make the number negative?
No. A negative exponent means take the reciprocal, so two to the power minus three is one-eighth, which is positive. The sign of the result depends only on the base. This confusion is the single most common error in the exponents topic.
What is standard form and what makes it correct?
It is writing a number as k times ten to the power n, where k is at least one and less than ten. So 59,000 is 5.9 times ten to the fourth. Writing it as 59 times ten cubed has the right value but is not standard form, and questions test exactly that.
How do I compare two numbers written in standard form?
Compare the powers of ten first, since a larger power means a larger number regardless of the coefficient. Only when the powers are equal do you compare the values of k. This is faster and safer than converting both numbers back to ordinary notation.
Next step: The number system is the base every other maths chapter stands on. Test it now in the BPSC TRE 4.0 mock test on ExamAtlas — the same five-option pattern with −1/3 negative marking, so you find the gaps before the exam does.