Solving Equations with Indices

In this topic, you'll learn how to solve equations where numbers are raised to a power. These powers are called indices (or exponents). For example, in the expression $2^3$ , the number $2$ is the base, and $3$ is the index, which tells us to multiply $2$ by itself three times: $2 \times 2 \times 2 = 8$ .

When solving equations with indices, your goal is to find the value of the unknown variable (often called $x$ that makes the equation true. To do this, we use some basic rules about how indices work. These rules will help you rewrite the equation in a way that makes it easier to solve.

Key Concepts:

Indices (Powers): The index (or exponent) tells you how many times to multiply a number by itself. For example, $2^3 = 2 \times 2 \times 2 = 8$ .
Basic Laws of Indices (Page 21 of the Formulae and Tables book):

$a^p \times a^q = a^{p+q}$
$\frac{a^p}{a^q} = a^{p-q}$
$(a^p)^q = a^{pq}$
$a^0 = 1$
$a^{-p} = \frac{1}{a^p}$
$a^{\frac{1}{q}} = \sqrt[q]{a}$
$a^{\frac{p}{q}} = \sqrt[q]{a^p} = (\sqrt[q]{a})^p$
$(ab)^p = a^p b^p$
$\left(\frac{a}{b}\right)^p = \frac{a^p}{b^p}$

Important Rule: If $a^x = a^y$ , then $x = y$ . Explanation: When the bases (the numbers being raised to a power) are the same, and the two sides of the equation are equal, it means that the powers (the numbers on top) must be equal. This is a key rule when solving equations with indices.

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Example: If $2^x = 2^3$ , then $x = 3$ .

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Worked Examples:

Problem: Solve the equation $2^x = 16.$

Step-by-Step Solution:

Express the right side as a power of $2$ : $16 = 2^4$ . Explanation: We express $16$ as $2^4$ because it helps us compare both sides of the equation directly. By having the same base (in this case, $2$ ), we can easily solve for $x$ .
Rewrite the equation: Now, the equation looks like this: $2^x = 2^4$ .
Apply the rule (if $a^x = a^y$ , then $x = y$ ): Since the bases are the same on both sides, the exponents (powers) must be equal. Therefore, $x = 4$ .
Final Answer: $x = 4$ .

Problem: Solve $3^{2x-1} = 27$ .

Step-by-Step Solution:

Express $27$ as a power of $3$ : $27 = 3^3$ . Explanation: We express $27$ as $3^3$ so that the bases on both sides of the equation are the same, making it easier to solve for $x$ .
Rewrite the equation: Now, the equation looks like this: $3^{2x-1} = 3^3$ .
Apply the rule (if $a^x = a^y$ , then $x = y$ ): Since the bases are the same, we can equate the exponents: $2x - 1 = 3$ .
Solve for $x$ : First, add $1$ to both sides: $2x = 4$ . Then, divide both sides by $2$ : $x = 2$ .
Final Answer: $x = 2$ .

Problem: Solve $5^x = \frac{1}{5}$ .

Step-by-Step Solution:

Rewrite the fraction as a negative power: $\frac{1}{5} = 5^{-1}.$ Explanation: We rewrite $\frac{1}{5}$ as $5^{-1}$ because it allows us to express both sides of the equation with the same base ( $5$ ).
Rewrite the equation: Now, the equation looks like this: $5^x = 5^{-1}.$
Apply the rule (if $a^x = a^y$ , then $x = y$ ): Since the bases are the same, the exponents must be equal. Therefore, $x = -1$ .
Final Answer: $x = -1$ .

Problem: Solve $2^{x+1} = 8$ .

Step-by-Step Solution:

Express $8$ as a power of $2$ : $8 = 2^3$ . Explanation: We express $8$ as $2^3$ so that the bases on both sides of the equation are the same, making it easier to solve for $x$ .
Rewrite the equation: Now, the equation looks like this: $2^{x+1} = 2^3.$
Apply the rule (if $a^x = a^y$ , then $x = y$ ): Since the bases are the same, the exponents must be equal: $x + 1 = 3$ .
Solve for $x$ : Subtract $1$ from both sides: $x = 2$ .
Final Answer: $x = 2.$

Problem: Solve $4^{2x} = 4$ .

Step-by-Step Solution:

Recognise that 4 is the same as $4^1$ : $4 = 4^1$ . Explanation: We recognise that $4$ can be written as $4^1$ so that both sides of the equation have the same base.
Rewrite the equation: Now, the equation looks like this: $4^{2x} = 4^1$ .
Apply the rule (if $a^x = a^y$ , then $x = y$ ): Since the bases are the same, the exponents must be equal: $2x = 1$ .
Solve for $x$ : Divide both sides by $2$ : $x = \frac{1}{2}$ .
Final Answer: $x = \frac{1}{2}$ .

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Exam Tips:

Simplify First: Always simplify both sides of the equation before solving. Factorizing or expressing numbers as powers of a common base can often make the problem easier to handle.
Apply the Rule: If the bases are the same, the indices must be equal. This rule is crucial for solving equations with indices.
Check Your Solutions: After solving, substitute your solution back into the original equation to ensure it works.
Practice: The more you practice, the more familiar you will become with these types of problems.

Solving Equations with Indices Simplified Revision Notes for Junior Cycle Mathematics

Solving Equations with Indices

Key Concepts:

Worked Examples:

Exam Tips:

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