Sandra was simplifying 45√×23√, and she ended up with 815−−√ as her simplified answer. Is Sandra correct? If she is not correct, what was her mistake?
To simplify the expression $45\sqrt{23}\cdot{}\sqrt{815}$, we first recognize that we can combine the square roots as follows: \begin{align*}
45\sqrt{23}\cdot{}\sqrt{815}&=45\sqrt{23\cdot815}\\
&=45\sqrt{18745}.
\end{align*} Instead of going further, we can conclude that Sandra made a mistake because the expression $\sqrt{815}-\sqrt{18745}$ cannot be simplified any further. Therefore, Sandra's answer is incorrect.
To determine if Sandra is correct, we can simplify the expression ourselves.
We start by simplifying each square root individually:
√45 = √(9 × 5) = √9 × √5 = 3√5
√23 = √(4 × 5 + 3) = √(4 × 5) + √3 = 2√5 + √3
Now, we multiply the two simplified square roots:
(3√5) × (2√5 + √3) = 3√5 × 2√5 + 3√5 × √3
= 6(√5)^2 + 3√5√3
= 6(5) + 3√15
= 30 + 3√15
Therefore, the simplified expression is 30 + 3√15, not 815-√.
Sandra made a mistake in her calculations. She might have made an error while multiplying the square roots or misunderstood the process of simplifying square roots.
To determine if Sandra's answer is correct, we need to simplify the expression 45√ × 23√ and compare it to 815−−√.
To simplify the given expression, we can use the following property of radicals: √(a × b) = √(a) × √(b).
Let's begin by simplifying 45√:
√45 = √(9 × 5) = √9 × √5 = 3√5
Next, we'll simplify 23√:
√23
Therefore, the simplified expression should be:
3√5 × √23
To multiply these expressions, we combine the constants and the radicals:
3√(5 × 23) = 3√115
Now, let's compare Sandra's answer, 815−−√, to the simplified expression, 3√115:
To find the value of 815−−√, we need to square it:
(815−−√)^2 = (815)^(2/2) = 815
Since 815 is not equal to 3√115, Sandra's answer is incorrect.
Therefore, her mistake lies in failing to multiply the two expressions when she simplified the original expression.