Solving Equations with Indices Revision Notes for Junior Cert Mathematics

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. Factorising 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 (Junior Cert Mathematics): Revision Notes