Certainly! Let's solve each part step-by-step.
### Part (a)
[tex]\[ 2^3 \times 2^5 \][/tex]
To solve this, use the properties of exponents. When you multiply two powers with the same base, you add the exponents:
[tex]\[ 2^3 \times 2^5 = 2^{3+5} = 2^8 \][/tex]
Now calculate [tex]\( 2^8 \)[/tex]:
[tex]\[ 2^8 = 256 \][/tex]
So, the value of [tex]\( 2^3 \times 2^5 \)[/tex] is [tex]\( 256 \)[/tex].
### Part (b)
[tex]\[ \left(\frac{3}{5}\right)^3 \][/tex]
Raise the fraction to the power of 3:
[tex]\[ \left(\frac{3}{5}\right)^3 = \frac{3^3}{5^3} = \frac{27}{125} \][/tex]
[tex]\[ \frac{27}{125} \approx 0.216 \][/tex]
So, the value of [tex]\( \left(\frac{3}{5}\right)^3 \)[/tex] is approximately [tex]\( 0.216 \)[/tex].
### Part (c)
[tex]\[ 10^6 \div 10^4 \][/tex]
To solve this, use the properties of exponents. When you divide two powers with the same base, you subtract the exponents:
[tex]\[ 10^6 \div 10^4 = 10^{6-4} = 10^2 \][/tex]
Now calculate [tex]\( 10^2 \)[/tex]:
[tex]\[ 10^2 = 100 \][/tex]
So, the value of [tex]\( 10^6 \div 10^4 \)[/tex] is [tex]\( 100 \)[/tex].
### Part (d)
[tex]\[ \left(\frac{1}{3}\right)^3 \times \left(\frac{3}{2}\right)^2 \][/tex]
Calculate each term separately:
[tex]\[ \left(\frac{1}{3}\right)^3 = \frac{1^3}{3^3} = \frac{1}{27} \][/tex]
[tex]\[ \left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4} \][/tex]
Now multiply the two fractions:
[tex]\[ \frac{1}{27} \times \frac{9}{4} = \frac{1 \times 9}{27 \times 4} = \frac{9}{108} = \frac{1}{12} \][/tex]
[tex]\[ \frac{1}{12} \approx 0.083 \][/tex]
So, the value of [tex]\( \left(\frac{1}{3}\right)^3 \times \left(\frac{3}{2}\right)^2 \)[/tex] is approximately [tex]\( 0.083 \)[/tex].
### Part (e)
[tex]\[ (-4)^3 \times (-1)^2 \times 2^2 \][/tex]
Calculate each term separately:
[tex]\[ (-4)^3 = -64 \][/tex]
[tex]\[ (-1)^2 = 1 \][/tex]
[tex]\[ 2^2 = 4 \][/tex]
Now multiply the results:
[tex]\[ -64 \times 1 \times 4 = -256 \][/tex]
So, the value of [tex]\( (-4)^3 \times (-1)^2 \times 2^2 \)[/tex] is [tex]\( -256 \)[/tex].
### Part (f)
[tex]\[ 2^3 + 3^2 + 4^3 \][/tex]
Calculate each term separately:
[tex]\[ 2^3 = 8 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 4^3 = 64 \][/tex]
Now add the results:
[tex]\[ 8 + 9 + 64 = 81 \][/tex]
So, the value of [tex]\( 2^3 + 3^2 + 4^3 \)[/tex] is [tex]\( 81 \)[/tex].
In conclusion, the values are:
- (a) [tex]\( 256 \)[/tex]
- (b) [tex]\( 0.216 \)[/tex]
- (c) [tex]\( 100 \)[/tex]
- (d) [tex]\( 0.083 \)[/tex]
- (e) [tex]\( -256 \)[/tex]
- (f) [tex]\( 81 \)[/tex]
