We are given a set of math problems. a) (i) Express 5630 in standard form. (ii) Express 16346000000 in engineering notation. b) Simplify the expression $(-5x^2y)(-2x^{-3}y^2)$. c) Use the remainder theorem to find the remainder when $2x^3 + x^2 - 7x - 6$ is divided by $x-2$. d) Solve the simultaneous equations $7x - 3y = 23$ and $2x - 4y = -8$. e) (i) Solve $\log_3 x = -2$. (ii) Solve $4x - 7(2-x) = 3x + 2$.

1. Problem Description

We are given a set of math problems.

a) (i) Express 5630 in standard form. (ii) Express 16346000000 in engineering notation.

b) Simplify the expression $(-5x^2y)(-2x^{-3}y^2)$.

c) Use the remainder theorem to find the remainder when $2x^3 + x^2 - 7x - 6$ is divided by $x-2$.

d) Solve the simultaneous equations $7x - 3y = 23$ and $2x - 4y = -8$.

e) (i) Solve $\log_3 x = -2$. (ii) Solve $4x - 7(2-x) = 3x + 2$.

2. Solution Steps

a) (i) Standard form is scientific notation where the exponent is adjusted such that only one non-zero digit is to the left of the decimal.

$5630 = 5.63 \times 10^3$

(ii) Engineering notation is scientific notation where the exponent is a multiple of

3. $16346000000 = 16.346 \times 10^9$

b) To simplify $(-5x^2y)(-2x^{-3}y^2)$, multiply the coefficients and add the exponents of like variables.

$(-5x^2y)(-2x^{-3}y^2) = (-5)(-2)x^{2+(-3)}y^{1+2} = 10x^{-1}y^3 = \frac{10y^3}{x}$

c) The Remainder Theorem states that if we divide a polynomial $P(x)$ by $(x-a)$, the remainder is $P(a)$. Here, $P(x) = 2x^3 + x^2 - 7x - 6$ and we divide by $x-2$, so $a=2$.

$P(2) = 2(2^3) + (2^2) - 7(2) - 6 = 2(8) + 4 - 14 - 6 = 16 + 4 - 14 - 6 = 20 - 20 = 0$.

The remainder is

0.

d) We have the system of equations

$7x - 3y = 23$

$2x - 4y = -8$

Multiply the first equation by 4 and the second equation by -3 to eliminate $y$:

$28x - 12y = 92$

$-6x + 12y = 24$

Add the equations:

$22x = 116$

$x = \frac{116}{22} = \frac{58}{11}$

Substitute $x = \frac{58}{11}$ into the second equation:

$2(\frac{58}{11}) - 4y = -8$

$\frac{116}{11} - 4y = -\frac{88}{11}$

$-4y = -\frac{88}{11} - \frac{116}{11}$

$-4y = -\frac{204}{11}$

$y = \frac{204}{44} = \frac{51}{11}$

So, $x = \frac{58}{11}$ and $y = \frac{51}{11}$.

e) (i) $\log_3 x = -2$. Converting to exponential form:

$x = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

(ii) $4x - 7(2-x) = 3x + 2$

$4x - 14 + 7x = 3x + 2$

$11x - 14 = 3x + 2$

$8x = 16$

$x = 2$

3. Final Answer

a) (i) $5.63 \times 10^3$ (ii) $16.346 \times 10^9$

b) $\frac{10y^3}{x}$

c) 0

d) $x = \frac{58}{11}, y = \frac{51}{11}$

e) (i) $x = \frac{1}{9}$ (ii) $x = 2$