What is the fundamental difference between $a^m \times a^n$ and $(a^m)^n$?

In $a^m \times a^n$, you are multiplying two separate powers with the same base, so you add the indices ($a^{m+n}$). In $(a^m)^n$, you are raising an existing power to a new power, so you multiply the indices ($a^{mn}$).

How does the treatment of coefficients differ from the treatment of indices in an expression like $(2x^3)^4$?

The coefficient (2) is raised to the power of 4 ($2^4 = 16$), while the index of the variable ($x^3$) is multiplied by the power of 4 ($3 \times 4 = 12$). The result is $16x^{12}$.

When solving an equation like $3^x = 9^5$, why is it necessary to change the base of 9?

Index laws only apply when bases are identical. By rewriting 9 as $3^2$, the equation becomes $3^x = (3^2)^5$, allowing you to use the power of a power law to equate the exponents.

What is the common mistake when evaluating a negative index like $5^{-2}$?

Students often mistakenly think the answer is a negative number (e.g., -25). In reality, a negative index indicates a reciprocal, so $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$.

Why can't you simplify $x^2 + x^3$ using index laws?

Index laws only exist for multiplication, division, and powers. There is no rule for adding terms with the same base but different indices; they are not 'like terms' and cannot be combined.

What error occurs if you apply the power of a product law to a sum, such as $(x + y)^2$?

The power of a product law $(ab)^n = a^n b^n$ only applies to multiplication. For a sum, you must expand the brackets (e.g., $(x+y)(x+y) = x^2 + 2xy + y^2$), as the power does not simply distribute over addition.

Define the meaning of a fractional index $a^{\frac{1}{n}}$.

A fractional index where the numerator is 1 represents the $n$-th root of the base. For example, $a^{\frac{1}{2}}$ is the square root of $a$, and $a^{\frac{1}{3}}$ is the cube root of $a$.

What is the rule for a power of a quotient $(\frac{a}{b})^n$?

The power is applied to both the numerator and the denominator separately, resulting in $\frac{a^n}{b^n}$. This ensures the entire ratio is scaled by the exponent.

State the law for negative fractional indices, $a^{-\frac{m}{n}}$.

It combines the reciprocal rule and the root/power rule: $a^{-\frac{m}{n}} = \frac{1}{\sqrt[n]{a^m}}$. You take the $n$-th root, raise it to the power $m$, and then take the reciprocal.

Why is $a^0$ always equal to 1 (for $a \neq 0$)?

Based on the division law, $\frac{a^n}{a^n} = a^{n-n} = a^0$. Since any non-zero number divided by itself is 1, $a^0$ must be 1 to keep the laws of indices consistent.

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Laws of indices

Laws of Indices

Summary

The laws of indices are a set of mathematical rules used to simplify and manipulate expressions involving powers. These laws provide a consistent framework for performing operations like multiplication, division, and exponentiation on terms with the same base, extending from positive integers to zero, negative, and fractional exponents.

1. Definition & Core Concepts

Diagram showing the anatomy of a power expression with the base 'a' and the index 'n'.

2. Fundamental Operations: Multiplication and Division

Multiplication Law: When multiplying terms with the same base, you add the indices: $a^m \times a^n = a^{m+n}$ . This works because you are combining two groups of repeated multiplications into one longer sequence.
Division Law: When dividing terms with the same base, you subtract the indices: $a^m \div a^n = a^{m-n}$ . This represents the process of 'canceling out' common factors from the numerator and denominator.
These laws apply to any real number base (except zero for division) and any rational index, including negatives and fractions.

3. Power Laws and Distribution

4. Zero, Negative, and Fractional Indices

5. Key Distinctions

6. Exam Strategy & Tips