Properties of powers
You already know that $2^3 \cdot 2^5 = 2^8$. The difficulty in 4th ESO is not the rule on its own, but recognising it inside a longer expression, with different bases that must be converted to a common one, with letters, or hidden inside an equation. Here you practise exactly that: spotting which property to apply and using it without losing the thread.
The four properties, all together
Express as a single power of base $5$ by applying index laws: $\frac{5^4 \cdot 5^{-2}}{5^3}$.
Express as a single power of base $5$ by applying index laws: $\frac{5^4 \cdot 5^{-2}}{5^3}$.
Reduce to a single power using the appropriate laws: $\left((-7)^3\right)^{-4} \cdot (-7)^5$.
Reduce to a single power using the appropriate laws: $\left((-7)^3\right)^{-4} \cdot (-7)^5$.
Same exponent, different bases
Write as a single power by applying the laws of operations with powers having the same exponent: $15^4 : 3^4 \cdot 2^4$.
Write as a single power by applying the laws of operations with powers having the same exponent: $15^4 : 3^4 \cdot 2^4$.
State whether the following statement is true or false, justifying your answer using index laws: "Double $2^8$ is $4^8$".
State whether the following statement is true or false, justifying your answer using index laws: "Double $2^8$ is $4^8$".
Simplifying algebraic expressions
With letters it works in exactly the same way; the result is usually required without negative exponents.
Simplify the following algebraic expression, leaving the result with positive exponents: $\frac{x^3 \cdot y^{-2}}{x^{-1} \cdot y^4}$.
Simplify the following algebraic expression, leaving the result with positive exponents: $\frac{x^3 \cdot y^{-2}}{x^{-1} \cdot y^4}$.
Fully simplify the following algebraic expression, using index laws and leaving the result without negative exponents:…
Fully simplify the following algebraic expression, using index laws and leaving the result without negative exponents: $\left( \frac{a^{-2} \cdot b^3}{a \cdot b^{-1}} \right)^{-2}$.
Simplify the following algebraic expression, where $x$ and $y$ are non-zero real numbers and $a, b$ are positive integers:…
Simplify the following algebraic expression, where $x$ and $y$ are non-zero real numbers and $a, b$ are positive integers:
$\left( \frac{x^{a-b} \cdot y^{b+a}}{x^{a+b} \cdot y^{b-a}} \right)^{-1}$
Apply the index laws for powers with the same base and express the final result using positive exponents only.
Comparing powers without a calculator
To compare two powers, the trick is almost always to write them with the same base.
Without using a calculator, determine which of the following quantities is larger: $3^{50}$ or $9^{20}$. Justify your answer by transforming one of the powers so that…
Without using a calculator, determine which of the following quantities is larger: $3^{50}$ or $9^{20}$. Justify your answer by transforming one of the powers so that both have the same base.
A student carried out the following algebraic working to simplify an expression:…
A student carried out the following algebraic working to simplify an expression:
$\frac{10^4 \cdot 5^{-2}}{2^4 \cdot 5^2} = \frac{(2 \cdot 5)^4 \cdot 5^{-2}}{2^4 \cdot 5^2} = \frac{2^4 \cdot 5^4 \cdot 5^{-2}}{2^4 \cdot 5^2} = 5^{4 - 2 - 2} = 5^0 = 1$
Identify whether there is an error in their reasoning, explain it in detail, and provide the correct result by strictly applying index laws.
Equations and applications
A rumour in a school spreads by doubling every hour the number of people who know it. If $1000$ people initially know it, express as a product with a power the number of…
A rumour in a school spreads by doubling every hour the number of people who know it. If $1000$ people initially know it, express as a product with a power the number of people who will know it after $12$ hours, and calculate its final value.
A sum of money is invested at an annual compound interest rate such that each year the money is multiplied by $1,05$. If the initial investment is $2000$ €, write the…
A sum of money is invested at an annual compound interest rate such that each year the money is multiplied by $1,05$. If the initial investment is $2000$ €, write the expression in index form that calculates the capital after $t$ years. How much money will there be after $4$ years?
Find the values of $x$ and $y$ that simultaneously satisfy the following equations, using index laws to transform the system into a system of linear equations:…
Find the values of $x$ and $y$ that simultaneously satisfy the following equations, using index laws to transform the system into a system of linear equations:
$2^x \cdot 4^y = 256$
$\frac{3^{2x}}{9^y} = 27$