GCSE Maths: Indices, Factorisation, Prime Factors, and HCF/LCM

Flashcards covering fractional indices, index equations, advanced factorization, prime factorization, and HCF/LCM from the lecture notes.

What does an index in the form $\frac{1}{n}$ represent for a given number $x$?

It represents the $n^{\text{th}}$ root of $x$ ($\sqrt[n]{x}$).

What is the value of $25^{\frac{1}{2}}$?

$$5$$

What is the value of $8^{\frac{1}{3}}$?

$$2$$

What is the value of $81^{\frac{1}{4}}$?

$$3$$

How is a expression with a general fractional index $x^{\frac{a}{b}}$ evaluated?

Take the $b^{\text{th}}$ root of $x$, then raise the result to the power $a$ ($(\sqrt[b]{x})^a$).

What is the value of $27^{\frac{2}{3}}$?

$$9$$

What is the value of $16^{\frac{3}{2}}$?

$$64$$

What is the result of evaluating $\left(\frac{27}{64}\right)^{\frac{2}{3}}$?

$$\frac{9}{16}$$

How do you simplify $(2b^8)^3$ using index laws?

Raise $2$ to the power $3$ to get $8$, and multiply the powers $8 \times 3 = 24$, giving $8b^{24}$.

What is the simplified form of $(16a^4b^6)^{\frac{1}{2}}$?

$$4a^2b^3$$

What are the solutions to the equation $(27x^2)^{\frac{1}{3}} = 75$?

$x = 125$ and $x = -125$

How do you solve the equation $5^{x^2} \times 25^{2x} = \frac{1}{125}$?

Rewrite terms as powers of $5$ to get $5^{x^2 + 4x} = 5^{-3}$, set exponents equal to form $x^2 + 4x + 3 = 0$, factorize as $(x + 1)(x + 3) = 0$, giving $x = -1$ and $x = -3$.

What is the value of $x$ in the equation $25^{4x} = \frac{5^{59} - 5^{57}}{120}$?

$$x = 7$$

Why must negative solutions be rejected when solving $x^{\frac{3}{2}} - 20x^{-\frac{1}{2}} = 8x^{\frac{1}{2}}$ for $x > 0$?

Because $x^{\frac{1}{2}}$ represents the square root of $x$, and real square roots of negative numbers cannot be taken.

What are the solutions to the equation $x^{\frac{5}{3}} + 3x^{\frac{2}{3}} = 10x^{-\frac{1}{3}}$?

$x = -5$ and $x = 2$

How is the expression $7(2x + 5)^4 + 2(2x + 5)^5$ factorized?

Factor out $(2x + 5)^4$ to obtain $(2x + 5)^4 [7 + 2(2x + 5)]$, which simplifies to $(2x + 5)^4 (2x + 17)$.

What substitution transforms an expression like $x^4 - 7x^2 + 10$ into a standard quadratic form?

Let $y = x^2$, which gives $y^2 = x^4$, transforming the expression into $y^2 - 7y + 10$.

What is the complete factorisation of $x^4 - 81$?

$$(x^2 + 9)(x + 3)(x - 3)$$

How do you simplify the algebraic fraction $\frac{5x - 10x^2}{4x^2 - 1}$?

Factorize the numerator as $-5x(2x - 1)$ and denominator as $(2x + 1)(2x - 1)$, then cancel $(2x - 1)$ to get $-\frac{5x}{2x + 1}$.

What is a prime number according to the transcript?

A number that has exactly $2$ factors ($1$ and itself).

What is $120$ written as a product of prime factors in index notation?

$$2^3 \times 3 \times 5$$

What is $700$ expressed as a product of prime factors in index form?

$$2^2 \times 5^2 \times 7$$

If $a = 2^4 \times 3 \times 11$ and $b = 18a$, what is the prime factorisation of $b$?

$$2^5 \times 3^3 \times 11$$

What is the highest common factor (HCF) of $12$ and $18$?

$$6$$

What is the lowest common multiple (LCM) of $4$, $6$, and $9$?

$$36$$

If Tom gets a haircut every $3$ weeks, Mark every $4$ weeks, and Travis every $5$ weeks, in how many weeks will they all get haircuts on the same day?

$60$ weeks (the lowest common multiple of $3$, $4$, and $5$).

If Christine has $18$ apples and $24$ oranges, what is the maximum number of baskets she can use to distribute both fruits equally without leftovers?

$6$ baskets (the highest common factor of $18$ and $24$).

Given prime factorizations $84 = 2^2 \times 3 \times 7$ and $280 = 2^3 \times 5 \times 7$, how is a Venn diagram used to find the HCF?

Place the common factors ($2$, $2$, and $7$) in the intersection and multiply them: $2 \times 2 \times 7 = 28$.

Using a Venn diagram containing prime factorizations of $84$ and $280$, how is the LCM determined?

Multiply all numbers present anywhere inside the Venn diagram: $3 \times 2 \times 2 \times 7 \times 2 \times 5 = 840$.

Given $a = 2^8 \times 5^4 \times 11$ and $b = 2^3 \times 5 \times 11^3$, what are the HCF and LCM of $a$ and $b$ in index form?

The HCF is $2^3 \times 5 \times 11$ and the LCM is $2^8 \times 5^4 \times 11^3$.