Question
Simplify.
$( \mathbf{a} )$ $\frac {32g^{16}}{16g^{8}}$
$( \mathbf{b} )$ $( 625k^8) ^{\frac {1}{4}}$
▶️Answer/Explanation
[(a)] $2g^8$ (final answer)
[(b)] $5k^2$ (final answer)
(a)
$
\frac{32g^{16}}{16g^8}
$
Dividing the coefficients
$
\frac{32}{16} = 2
$
dividing powers
$
\frac{g^{16}}{g^8} = g^{16-8} = g^8
$
$
2g^8
$
(b)
$(625k^8)^{\frac{1}{4}}$
$
= 625^{\frac{1}{4}} \times (k^8)^{\frac{1}{4}}
$
$
625^{\frac{1}{4}} = 5
$
$
(k^8)^{\frac{1}{4}} = k^{8 \times \frac{1}{4}} = k^2
$
$
5k^2
$
Question
Factorise completely.
$4x^2y-5xy^2$
▶️Answer/Explanation
$xy(4x – 5y)$ final answer
Both terms have a common factor of \( xy \). So
$
= xy(4x – 5y)
$
The fully factorised expression is
$
\mathbf{xy(4x – 5y)}
$
Question
Calculate \(\sqrt{17.8}-1.3^{2.5}\)
Answer/Explanation
2.29 or 2.292
Question
(a) \(3^x = \sqrt[4]{3^5}\)
Find the value of x.
(b) Simplify \((32y^{15})^{\frac{2}{5}}\)
Answer/Explanation
Ans:
(a) \(\frac{5}{4}\) oe
(b) \(4y^6\)
Question
Simplify
(a) $\left(\frac{p^4}{16}\right)^{0.75}$,
(b) $3^2 q^{-3} \div 2^3 q^{-2}$.
▶️Answer/Explanation
(a)$\left(\frac{p^4}{16}\right)^{0.75} = \frac{p^{4 \times 0.75}}{16^{0.75}}$
Simplifying the exponents:
$\frac{p^3}{16^{0.75}}$
To simplify further, we can evaluate the exponent $16^{0.75}$:
$16^{0.75} = \sqrt[4]{16^3} = \sqrt[4]{4096} = 8$
Substituting this back into the expression:
$\frac{p^3}{8}$
Therefore, $\left(\frac{p^4}{16}\right)^{0.75}$ simplifies to $\frac{p^3}{8}$.
(b) To simplify $3^2 q^{-3} \div 2^3 q^{-2}$, we can simplify each term separately and then perform the division:
$3^2 = 9$
$q^{-3} = \frac{1}{q^3}$
$2^3 = 8$
$q^{-2} = \frac{1}{q^2}$
Substituting these simplified terms back into the expression:
$9 \cdot \frac{1}{q^3} \div 8 \cdot \frac{1}{q^2}$
Applying the division rule for exponents (subtracting exponents when dividing):
$\frac{9}{8} \cdot \frac{1}{q^3} \cdot q^2$
Simplifying:
$\frac{9}{8q}$
Therefore, $3^2 q^{-3} \div 2^3 q^{-2}$ simplifies to $\frac{9}{8q}$.