Practice Problems (Junior Cert Mathematics): Revision Notes

Practice Problems

Problems:

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Problem 1:

Explanation:

Use the first law of indices, which states that when you multiply powers with the same base, you add the exponents.

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Problem 2:

Explanation:

Use the second law of indices, which states that when you divide powers with the same base, you subtract the exponents.

Problem 3:

Explanation:

Use the third law of indices, which states that when you raise a power to another power, you multiply the exponents.

Problem 4:

Explanation:

Express $64$ as a power of $4$, then set the exponents equal to each other since the bases are the same.

Problem 5:

Explanation:

Express $16$ as a power of $2$, then solve for $x$ by setting the exponents equal to each other.

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Solutions:

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Problem 1:

• Step 1: Apply the first law of indices: $a^m \times a^n = a^{m+n}$.
  • $2^3 \times 2^4 = 2^{3+4}$
  • Explanation: We add the exponents because the base $(2)$ is the same. When you multiply numbers with the same base, you add their exponents.
• Step 2: Simplify the exponent.
  • $2^{3+4} = 2^7$
  • Explanation: The exponent simplifies to $7$, so the answer is $2^7$.
• Final Answer: $2^3 \times 2^4 = 2^7$

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Problem 2:

Step 1: Apply the second law of indices: $\frac{a^m}{a^n} = a^{m-n}$.

• $\frac{5^6}{5^2} = 5^{6-2}$
• Explanation: We subtract the exponents because the base $(5)$ is the same. When you divide numbers with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
• Step 2: Simplify the exponent.
• $5^{6-2} = 5^4$
• Explanation: The exponent simplifies to $4$, so the answer is $5^4$.
• Final Answer: $\frac{5^6}{5^2} = 5^4$

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Problem 3:

Step 1: Apply the third law of indices: $(a^m)^n = a^{m \times n}$.

• $(3^2)^3 = 3^{2 \times 3}$
• Explanation: We multiply the exponents because we are raising a power to another power.
• Step 2: Simplify the exponent.
• $3^{2 \times 3} = 3^6$
• Explanation: The exponent simplifies to $6$, so the answer is $3^6$.
• Final Answer: $(3^2)^3 = 3^6$

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Problem 4:

• Step 1: Express $64$ as a power of $4$.
  • $64 = 4^3$
  • Explanation: Recognise that 64 can be written as $4^3$ because $4 \times 4 \times 4 = 64$.
• Step 2: Set the exponents equal to each other since the bases are the same.
  • $4^x = 4^3$ implies $x = 3$
  • Explanation: Since the bases are the same, you can set the exponents equal to each other. This is based on the rule: If $a^m = a^n$, then $m = n$. This means if two expressions with the same base are equal, their exponents must also be equal.
• Final Answer: $x = 3$

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Problem 5:

• Step 1: Express $16$ as a power of $2$.
  • $16 = 2^4$
  • Explanation: Recognise that 16 can be written as $2^4$ because $2 \times 2 \times 2 \times 2 = 16$.
• Step 2: Set the exponents equal to each other.
  • $2^{x+1} = 2^4$ implies $x + 1 = 4$
  • Explanation: Since the bases are the same, you set the exponents equal to each other.
• Step 3: Solve for $x$.
  • $x = 4 - 1 = 3$
  • Explanation: Subtract $1$ from both sides to find the value of $x$.
• Final Answer: $x = 3$

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