Grade 8 Mathematics: Powers, Algebra, and Percentages Revision

Comprehensive practice flashcards covering Unit 1.3 (Powers, Roots, and Standard Form), Unit 2 (Algebra: Expressions, Single/Double Bracket Expansion, Factorisation, and Formulae), and Unit 4 (Fractions, Decimals, and Percentages) based on the Grade 8C Mathematics exam revision worksheet.

According to the index rules in Lesson 1.3a, what is any non-zero number raised to the power of 0, and what is any number raised to the power of 1?

Any number to the power of 1 is itself (for example, $3^1 = 3$), and any non-zero number to the power of 0 is 1 (for example, $7^0 = 1$).

How does a negative base behave when raised to an even power versus an odd power?

A negative base raised to an even power results in a positive value (for example, $(-3)^4 = 81$), whereas a negative base raised to an odd power results in a negative value (for example, $(-2)^3 = -8$).

In Lesson 1.3a, Rob states that $\sqrt[5]{-32} = \pm 2$. What mistake did Rob make, and what is the correct value?

Rob incorrectly applied the rule for square roots (which have both positive and negative roots) to an odd root. An odd root of a negative number has only one real solution because $(-2)^5 = -32$ while $2^5 = +32$. The correct answer is $-2$.

A cardboard cube has a volume of $15{,}625\,\text{cm}^3$. How do you calculate its surface area?

First, find the edge length by taking the cube root of the volume: $s = \sqrt[3]{15{,}625} = 25\,\text{cm}$. Then, calculate the total surface area of the 6 square faces: $\text{Surface Area} = 6 \times s^2 = 6 \times 25^2 = 6 \times 625 = 3750\,\text{cm}^2$.

What is the general form and condition for a number written in standard form?

A number in standard form is written as $A \times 10^n$, where $1 \le A < 10$ and $n$ is an integer.

Chloe writes $45{,}000{,}000$ in standard form as $45 \times 10^6$. What mistake did she make, and what is the correct answer?

Chloe did not ensure that the first number $A$ satisfies $1 \le A < 10$ (since $45 \ge 10$). The decimal point must sit immediately after the first non-zero digit. The correct standard form is $4.5 \times 10^7$.

What is the correct ascending order for the numbers: $4.2 \times 10^6$, $3.9 \times 10^{11}$, $4.11 \times 10^5$, $3.2 \times 10^7$, $4.09 \times 10^6$, and $5.2 \times 10^3$?

$5.2 \times 10^3$, $4.11 \times 10^5$, $4.09 \times 10^6$, $4.2 \times 10^6$, $3.2 \times 10^7$, $3.9 \times 10^{11}$.

In 2010, the populations of five countries were: China ($1.34 \times 10^9$), India ($1.18 \times 10^9$), United States ($3.09 \times 10^8$), Indonesia ($2.38 \times 10^8$), and Brazil ($1.93 \times 10^8$). What was their total population in standard form?

Summing the populations: $(1.34 \times 10^9 + 1.18 \times 10^9) + (3.09 \times 10^8 + 2.38 \times 10^8 + 1.93 \times 10^8) = 2.52 \times 10^9 + 7.40 \times 10^8 = 2.52 \times 10^9 + 0.74 \times 10^9 = 3.26 \times 10^9$.

If the world population in 2010 was $6.84 \times 10^9$ and the combined population of five countries was $3.26 \times 10^9$, what percentage of the world population lived in these five countries (to 1 decimal place)?

$\frac{3.26 \times 10^9}{6.84 \times 10^9} \times 100\% \approx 47.7\%$.

For the algebraic expression $2x^2 - 3x + 9$, identify the number of terms, the coefficient of $x^2$, the coefficient of $x$, and the constant term.

Number of terms: $3$ Coefficient of $x^2$: $2$ Coefficient of $x$: $-3$ Constant term: $9$.

Evaluate $2bc - 3a^2$ when $a = 4$, $b = 2$, and $c = 5$.

$2(2)(5) - 3(4^2) = 20 - 3(16) = 20 - 48 = -28$.

Rowan claims that when $a = -2$ and $b = 3$, the value of $\frac{a^2 b}{3a^3}$ is $\frac{1}{2}$. What error did Rowan make, and what is the correct value?

Rowan failed to include the negative sign resulting from cubing a negative base in the denominator (or simplifying first: $\frac{a^2 b}{3a^3} = \frac{b}{3a} = \frac{3}{3(-2)} = -\frac{1}{2}$). Evaluating directly gives $\frac{(-2)^2 (3)}{3(-2)^3} = \frac{12}{-24} = -\frac{1}{2}$.

Write an algebraic expression for: 'There are $c$ cookies in a box. Three friends share them equally. Eliza takes her share, then gives 2 cookies to her sister.'

$\frac{c}{3} - 2$.

Formulate and solve an equation for: 'I think of a number $x$, add 4, and then multiply by 3. My answer is 57.'

Equation: $3(x + 4) = 57$. Divide by 3: $x + 4 = 19$. Subtract 4: $x = 15$.

Expand and simplify the algebraic expression $3(x + y) - 2(x - y)$.

$3x + 3y - 2x + 2y = x + 5y$.

Rowan expanded $xy(x - y)$ to $x^2 y + xy^2$. Explain what he did wrong and state the correct expansion.

Rowan forgot that multiplying a positive term by a negative term produces a negative term ($xy \times (-y) = -xy^2$, not $+xy^2$). The correct expansion is $x^2 y - xy^2$.

Fred says that the result of factorising $3x^5 y + 12x^4 y^2$ is $3x^4(xy + 4y^2)$. What mistake did Fred make, and what is the fully factorised form?

Fred did not factor out the common variable $y$ present in both terms inside the bracket. The highest common factor is $3x^4 y$, giving the fully factorised expression $3x^4 y(x + 4y)$.

Factorise completely: $18x^5 + 9x^6 - 36x^8$.

The highest common factor is $9x^5$. Factoring it out gives $9x^5(2 + x - 4x^3)$.

Rob factorises $x^3 y^3 z - x^2 y^2 z$ as $xy^2 z(x^2 y - x)$. How can you tell from the terms inside the bracket that he has not factorised completely, and what is the correct factorisation?

Both terms inside the bracket $(x^2 y - x)$ still share a common factor of $x$. Taking out the full HCF of $x^2 y^2 z$ gives the correct factorisation: $x^2 y^2 z(xy - 1)$.

Helen claims that $(x + 6)^2$ is equivalent to $x^2 + 36$. Explain why she is incorrect.

Squaring a binomial requires multiplying it by itself: $(x + 6)(x + 6) = x^2 + 6x + 6x + 36 = x^2 + 12x + 36$. Helen omitted the two middle cross-multiplication terms ($6x + 6x = 12x$).

Show that $(x + 2)(x - 9) - 5(x - 4) \equiv x^2 - 12x + 2$.

Expand the double brackets: $(x + 2)(x - 9) = x^2 - 9x + 2x - 18 = x^2 - 7x - 18$. Expand the single bracket: $-5(x - 4) = -5x + 20$. Combine like terms: $x^2 - 7x - 18 - 5x + 20 = x^2 - 12x + 2$.

Make $t$ the subject of the scientific formula $v = u + at$.

Subtract $u$ from both sides: $v - u = at$. Divide both sides by $a$: $t = \frac{v - u}{a}$.

Rearrange the area formula of a triangle, $A = \frac{1}{2}bh$, to make $b$ the subject, then find $b$ when $A = 12\,\text{m}^2$ and $h = 2.5\,\text{m}$.

Multiply by 2 and divide by $h$: $b = \frac{2A}{h}$. Substitute the values: $b = \frac{2(12)}{2.5} = \frac{24}{2.5} = 9.6\,\text{m}$.

Fred says that making $a$ the subject of $\frac{a+b}{c} = d$ gives $a = d - \frac{b}{c}$. Explain his mistake and give the correct formula.

Fred subtracted $b$ before undoing the division by $c$, failing to multiply $d$ by $c$. The correct method is to multiply both sides by $c$ first: $a + b = cd$, then subtract $b$: $a = cd - b$.

A plumber charges according to $P = 35t + 50$, where $P$ is the total price in dollars and $t$ is the time in hours. Identify the hourly rate and the total cost for a 3-hour job.

Hourly rate: \n$35\,\text{per hour} Cost of a 3-hour job: $P = 35(3) + 50 = 105 + 50 = \$155$.

The cost of hiring a restaurant for a party is $\$30$ per person plus a flat fee of $\$50$. If Shannon has $\$400$ and must pay for herself, how many people can she invite?

Formula: $C = 30n + 50$. Set inequality: $30n + 50 \le 400 \Rightarrow 30n \le 350 \Rightarrow n \le 11.67$. The maximum total number of attendees is 11. Since Shannon counts as one person, she can invite $11 - 1 = 10$ people.

Program A has $\frac{1}{3}$ left to download, while Program B has $0.35$ left to download. Which program is closer to being fully downloaded, and why?

Program A is closer because $\frac{1}{3} = 0.333\dots$, which is less than $0.35$. A smaller remaining amount means Program A has downloaded more.

A car was priced at $\$18{,}000$ in 2020. If its price increased by $12\%$ in 2021, calculate the increase amount and the new price in 2021.

Increase amount: $0.12 \times 18{,}000 = \$2160$. New price: $18{,}000 + 2160 = \$20{,}160$.

What single decimal multiplier represents: (a) an increase of $19\%$, (b) a decrease of $8\%$, and (c) a decrease of $3.5\%$?

(a) Increase of $19\%$: $1 + 0.19 = 1.19$ (b) Decrease of $8\%$: $1 - 0.08 = 0.92$ (c) Decrease of $3.5\%$: $1 - 0.035 = 0.965$.

Emily's train ticket increases from 20 dirhams to 25 dirhams. She calculates this as a $20\%$ increase. What mistake did Emily make, and what is the correct percentage increase?

Emily divided the change of 5 dirhams by the new price ($25$) instead of the original price ($20$): $\frac{5}{25} \times 100\% = 20\%$. The correct percentage increase is $\frac{25 - 20}{20} \times 100\% = \frac{5}{20} \times 100\% = 25\%$.