Indices and Scientific Notation

Only $a^m \times a^n = a^{m+n}$ is a genuine index law. The traps: powers cannot be added across a sum, $(a^m)^n = a^{mn}$ (multiply), $a^m \times b^m = (ab)^m$, and $a^0 = 1$.

$3^4 = 81$ and $4^3 = 64$, so $3^4$ is larger. (A larger base does not always give a larger value; neither equals $3 \times 4 = 12$.)

By order of operations the exponent applies only to the 3, not the negative sign: $3^2 = 9$ first, then the negative gives $-9$. (If it were $(-3)^2$, the negative is inside the base: $(-3)\times(-3)=9$. The brackets are the trap.)

Multiplying powers with the same base, add the exponents: $x^5\times x^3 = x^{5+3} = x^8$. (Multiplying the exponents is the power-of-a-power rule, not this one.)

Dividing powers with the same base, subtract the exponents: $\dfrac{y^7}{y^2} = y^{7-2} = y^5$. (Adding gives $y^9$; subtracting in the wrong order gives $y^{-5}=\dfrac{1}{y^5}$.)

Raising a power to a power, multiply the exponents: $(a^2)^3 = a^{2\times 3} = a^6$. (Adding gives $a^5$.)

Anything (non-zero) to the power 0 is 1: $x^0 = 1$ so $5x^0 = 5$, and $(3x)^0 = 1$. So $5 + 1 = 6$. ($5x$ wrongly kept the $x$; the whole $(3x)^0$ is 1, not 3.)

The side of a square is the square root of its area: $\sqrt{144} = 12$ cm (since $12^2 = 144$). ($72$ cm halved the area instead of taking the square root.)

The edge of a cube is the cube root of its volume: $\sqrt[3]{64} = 4$ cm (since $4^3 = 64$). ($8$ cm is the square root; $21.3$ cm divided by 3 instead of taking the cube root.)

Apply the outer exponent to each factor: $(3x^2)^3 = 3^3\times (x^2)^3 = 27x^6$. ($9x^6$ uses $3^2$; $3x^6$ forgets the coefficient.)

A negative exponent means the reciprocal: $2^{-2} = \dfrac{1}{2^2} = \dfrac{1}{4}$. ($-4$ treats the exponent as a factor; 4 ignores the negative.)

$\sqrt{81}$ is the number that squares to 81. Since $9\times 9 = 81$, $\sqrt{81}=9$. (40.5 halves 81; $6561 = 81^2$.)

$\sqrt[3]{27}$ is the number that cubes to 27. Since $3\times 3\times 3 = 27$, $\sqrt[3]{27}=3$. (9 comes from $\dfrac{27}{3}$; $19{,}683 = 27^3$.)

Find the perfect squares around 50: $7^2 = 49$ and $8^2 = 64$. Since $49 < 50 < 64$, $7 < \sqrt{50} < 8$ (about 7.07). (25 and 36 are the perfect squares, not their roots.)

Move the decimal so the first number is between 1 and 10: 6,400,000 becomes 6.4, and the point moved 6 places, so $6.4\times 10^6$. ($64\times 10^5$ has a first number above 10.)

Multiply: $3\times 10^4 \times 500 = 1500\times 10^4 = 1.5\times 10^7$ items. ($15\times 10^6$ is the right value, but the first number must be between 1 and 10.)

For a number less than 1 the exponent is negative. Moving the decimal 4 places gives 2.7, so $0.00027 = 2.7\times 10^{-4}$. ($2.7\times 10^4$ forgets the negative.)

A negative exponent means a small number: divide by $10^3 = 1000$, so $\dfrac{4.5}{1000} = 0.0045$ (move the point 3 places left). (4500 moves it the wrong way.)

Valid scientific notation needs a first number between 1 and 10, a base of 10, and multiplication. Only $5.2\times 10^7$ meets all three. ($52\times 10^6$ is above 10; $0.52\times 10^8$ is below 1.)

Compare the exponents first. $10^5$ beats $10^4$ and $10^3$, so $1.2\times 10^5 = 120{,}000$ is the largest. ($9.2\times 10^3 = 9200$ has the biggest first number but the smallest exponent.)

Multiply the first numbers and add the exponents: $(2\times 10^3)\times(3\times 10^4) = 6\times 10^{3+4} = 6\times 10^7$. ($6\times 10^{12}$ multiplies the exponents.)

By Pythagoras, $h^2 = 6^2 + 8^2 = 36 + 64 = 100$, so $h = \sqrt{100} = 10$ cm. ($14$ cm added the two sides; $100$ cm forgot to take the square root.)

Multiply: $6.5\times 10^2 \times 4 = 26\times 10^2 = 2600$, then write in scientific notation: $2.6\times 10^3$. ($26\times 10^2$ is the right value but the first number must be between 1 and 10.)

Apply the outer power: $(2x^3y^2)^3 = 2^3 x^9 y^6 = 8x^9y^6$. Then divide by $4x^5y^3$: coefficients $\dfrac{8}{4} = 2$, and subtract exponents to get $x^4$ and $y^3$. So $2x^4y^3$. ($8x^4y^3$ forgets to divide the coefficient.)

Time $= \dfrac{\text{distance}}{\text{speed}} = \dfrac{1.5\times 10^8}{3\times 10^5} = 0.5\times 10^3 = 500$ seconds. (Divide the numbers and subtract the powers of ten separately.)