Exponents

Using the power-of-a-power rule \\( (a^m)^n=a^{mn} \\), we get \\( (3^3)^3=3^{3\times3}=3^9 \\). Since \\(3^9=19683\\), the correct answer is 19683. The closest distractor, 6561, equals \\(3^8\\), which would result from an incorrect exponent calculation. Exam tip: when a power is raised to another power, multiply the exponents rather than adding them.

Using the exponent rule \(a^n \div b^n = (a \div b)^n\), we get \(18^2 \div 3^2 = (18 \div 3)^2 = 6^2 = 36\). Option 6 is only the quotient of the bases and does not include the square. In an exam, when the powers are equal, divide the bases first and then apply the common power.

Evaluate the squares: \(15^2=225\) and \(12^2=144\). Therefore, \(15^2-12^2=225-144=81\). The result can also be checked using the difference-of-squares identity \(a^2-b^2=(a-b)(a+b)\): \((15-12)(15+12)=3\times27=81\). Exam tip: use the difference-of-squares identity to calculate such expressions quickly and reduce arithmetic errors.

Here, 19^2=361 and 11^2=121, so 19^2−11^2=361−121=240. It can also be checked using the difference of squares: a^2−b^2=(a−b)(a+b)=(19−11)(19+11)=8×30=240. In an exam, use this factorisation to avoid calculating large squares directly.

When powers with the same base are multiplied, their exponents are added, so \(a^m\times a^n=a^{m+n}\). In option C, exponents multiply: \((a^m)^n=a^{mn}\). Exam tip: first check that the bases are identical.

Since 27 = 3^3, we have 27^2 = (3^3)^2 = 3^6. Therefore, 27^2 \div 3^4 = 3^6 \div 3^4 = 3^{6-4} = 3^2 = 9. When powers with the same base are divided, their exponents are subtracted; hence the nearby option 27 is incorrect. Exam tip: subtract the exponents when dividing powers with the same base.

Using the power-of-a-power rule, ((a^m)^n=a^{mn}), we get (2^5)^2=2^{10}. Then, by the quotient rule for equal bases, (2^m\div2^n=2^{m-n}), so 2^{10}\div2^6=2^{10-6}=2^4=16. Therefore, 16 is correct. Exam tip: when dividing powers with the same base, subtract the exponents; do not add them.

First evaluate the powers: \(2^6=64\) and \(3^5=243\). Adding them gives \(64+243=307\), so option A is correct. Remember to evaluate the exponents before adding the terms.

Evaluate the powers first: 14^2 = 196 and 5^3 = 125. Therefore, 14^2 + 5^3 = 196 + 125 = 321, so option B is correct. Exam tip: calculate exponents before addition; do not confuse 14^2 with 14 × 2.

When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Therefore, \(3^8 \div 3^5=3^{8-5}=3^3=27\). Hence, 27 is correct. Exam tip: for division with the same base, subtract the exponents; do not multiply them.

For a negative exponent, \(a^{-n}=\frac{1}{a^n}\), provided \(a\neq0\). Thus, \(0^{-3}=\frac{1}{0^3}\), which involves division by zero and is not defined. An expression such as \((-5)^{-2}\) is defined because its base is non-zero. Exam tip: Whenever you see a negative exponent, first check whether the base is zero.

A negative exponent represents the reciprocal of the corresponding positive power. Hence, \(x^{-m}=\frac{1}{x^m}\), provided \(x\neq0\). Option B incorrectly treats a negative exponent as merely adding a minus sign. Also, \(x^m\times x^{-m}=x^0=1\). Exam tip: For multiplication with the same base, add the exponents; rewriting negative powers as reciprocals helps avoid sign errors.

First evaluate the powers: 8^2 = 8 × 8 = 64 and 7^3 = 7 × 7 × 7 = 343. Therefore, 8^2 + 7^3 = 64 + 343 = 407, so option C is correct. Remember to evaluate exponents before performing the addition.

Rewrite 9 as a power of 3: 9 = 3^2. Thus, 9^3 = (3^2)^3 = 3^6. Using the quotient rule for powers with the same base, 3^6 \div 3^5 = 3^{6-5} = 3. Therefore, the correct answer is 3. Exam tip: when dividing powers with the same non-zero base, subtract the exponents: a^m \div a^n = a^{m-n}.

Evaluate the powers first: \(12^2=144\) and \(5^2=25\). Thus, \(12^2+5^2=144+25=169\). Finally, \(169\div13=13\), so option C is correct. Exam tip: simplify the exponents before carrying out the addition and division; treating \(12^2\) as \(12+2\) would be incorrect.

Evaluate the powers first: \(2^7=128\), \(3^4=81\), and \(5^2=25\). Then \(128+81-25=209-25=184\), so option B is correct. Values such as 174 or 194 usually result from an arithmetic error during addition or subtraction. Exam tip: evaluate exponents before carrying out addition and subtraction.

For powers with the same base, exponents are added during multiplication and subtracted during division: \(2^8 \times 2^5 \div 2^9=2^{8+5-9}=2^4=16\). Therefore, the correct answer is 16. The option 8 equals only \(2^3\), so it is incorrect. Exam tip: for the same base, add exponents when multiplying and subtract them when dividing.

First apply the power-of-a-power rule: (a^m)^n = a^{mn}, so (3^2)^4 = 3^8. Then use the quotient rule for powers with the same base: 3^8 \div 3^5 = 3^{8-5} = 3^3 = 27. Therefore, option B is correct. Exam tip: multiply exponents for a power raised to a power, but subtract exponents when dividing like bases.

A negative exponent represents a reciprocal. Thus, \((ab)^{-n}=\frac{1}{(ab)^n}=\frac{1}{a^n b^n}=a^{-n}b^{-n}\). Option B incorrectly uses addition; there is no exponent law that turns the reciprocal of a product into a sum of reciprocals. Exam tip: When an exponent is applied to a product in brackets, first apply the power-of-a-product rule.

A negative exponent does not make the base or the result negative; it denotes the reciprocal of the corresponding positive power. Therefore, \(a^{-n}=\dfrac{1}{a^n}\) for \(a\ne0\). Option B wrongly interprets the negative exponent as a minus sign, while option D is true only in some special cases, not generally. Exam tip: Rewrite every negative exponent as a reciprocal before simplifying.