Identify the greater number in each of the following— (i) or (ii) or (iii) or

Understand the Question An exponential expression represents the base multiplied by itself times. To find which number is greater, evaluate each expression numerically or compare them by expressing powers using exponent laws such as . Part (i) or Step 1 · Evaluate and Compare and Evaluate Evaluate Comparing the values Answer > (i) Part (ii) or Step 1 · Evaluate and Compare and Evaluate Evaluate Comparing the values Answer > (ii) Part (iii) or Step 1 · Evaluate and Compare with Evaluate Rewrite using the rule Calculate Substitute back into the expression Since , is far greater than . Therefore Answer > (iii) Common Mistakes Base Exponent Error: Incorrectly multiplying the base by the power (e.g., calculating as instead of ). Base Size Assumption: Incorrectly assuming a larger base always yields a larger number (e.g., thinking because , ignoring the massive exponent ).

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Question 11

Identify the greater number in each of the following—

(i) $4^3$ or $3^4$

(ii) $2^8$ or $8^2$

(iii) $100^2$ or $2^{100}$

Solution

Understand the Question

An exponential expression $a^b$ represents the base $a$ multiplied by itself $b$ times.
To find which number is greater, evaluate each expression numerically or compare them by expressing powers using exponent laws such as $(a^m)^n = a^{m \times n}$ .

(i) $4^3$ or $3^4$

Step 1 · Evaluate and Compare $4^3$ and $3^4$

Evaluate $4^3$

\begin{aligned} 4^3 &= 4 \times 4 \times 4 \\ &= 16 \times 4 \\ &= 64 \end{aligned}

Evaluate $3^4$

\begin{aligned} 3^4 &= 3 \times 3 \times 3 \times 3 \\ &= 9 \times 3 \times 3 \\ &= 27 \times 3 \\ &= 81 \end{aligned}

Comparing the values $81 > 64 \implies 3^4 > 4^3$

Answer

(i) $3^4$

(ii) $2^8$ or $8^2$

Step 1 · Evaluate and Compare $2^8$ and $8^2$

Evaluate $2^8$

\begin{aligned} 2^8 &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ &= 4 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ &= 8 \times 2 \times 2 \times 2 \times 2 \times 2 \\ &= 16 \times 2 \times 2 \times 2 \times 2 \\ &= 32 \times 2 \times 2 \times 2 \\ &= 64 \times 2 \times 2 \\ &= 128 \times 2 \\ &= 256 \end{aligned}

Evaluate $8^2$

\begin{aligned} 8^2 &= 8 \times 8 \\ &= 64 \end{aligned}

Comparing the values $256 > 64 \implies 2^8 > 8^2$

Answer

(ii) $2^8$

(iii) $100^2$ or $2^{100}$

Step 1 · Evaluate $100^2$ and Compare with $2^{100}$

Evaluate $100^2$

\begin{aligned} 100^2 &= 100 \times 100 \\ &= 10000 \end{aligned}

Rewrite $2^{100}$ using the rule $(a^m)^n = a^{m \times n}$ $2^{100} = (2^{10})^{10}$

Calculate $2^{10}$

\begin{aligned} 2^{10} &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ &= 1024 \end{aligned}

Substitute back into the expression $2^{100} = (1024)^{10}$

Since $1024^2 = 1024 \times 1024 = 1048576 > 10000$ , $(1024)^{10}$ is far greater than $10000$ .

Therefore $2^{100} > 100^2$

Answer

(iii) $2^{100}$

Common Mistakes

Base $\times$ Exponent Error: Incorrectly multiplying the base by the power (e.g., calculating $4^3$ as $4 \times 3 = 12$ instead of $4 \times 4 \times 4 = 64$ ).
Base Size Assumption: Incorrectly assuming a larger base always yields a larger number (e.g., thinking $100^2 > 2^{100}$ because $100 > 2$ , ignoring the massive exponent $100$ ).

Question 11

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