Example Question - perfect squares
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Simplified square root of 125
La imagen muestra la raíz cuadrada de 125. Para resolverla, buscaremos factores cuadrados perfectos para simplificar la expresión. 125 se puede descomponer en factores primos de la siguiente manera: 125 = 5 × 5 × 5 = \(5^2\) × 5 Ahora, usando las propiedades de las raíces, podemos separar la raíz cuadrada de \(5^2\) y la raíz cuadrada de 5: \(\sqrt{125}\) = \(\sqrt{5^2 \times 5}\) = \(\sqrt{5^2}\) × \(\sqrt{5}\) = 5 × \(\sqrt{5}\) Por lo tanto, la raíz cuadrada de 125 simplificada es 5\(\sqrt{5}\).
Finding Consecutive Whole Numbers for a Square Root Inequality
The image contains an inequality with a square root expression and two boxes. The inequality shows a square root of 83 \(\sqrt{83}\), and you are asked to find two consecutive whole numbers between which this value lies. To solve this, note that the value will be between the square roots of perfect squares that are closest to 83 but on either side of it. Since \(9^2 = 81\) and \(10^2 = 100\), \(\sqrt{83}\) will lie between 9 and 10. Hence, the solution to the inequality is: 9 < \(\sqrt{83}\) < 10 So, the two boxes should be filled with 9 and 10, respectively.
Solving Inequality with Square Roots
The image shows an inequality with two blank boxes and the square root of 136 in the center. To solve this, we need to find two square numbers that the square root of 136 falls between. Let's identify the nearest perfect squares around 136. Since \( 11^2 = 121 \) and \( 12^2 = 144 \), we can see that 136 falls between these two squares. So, the square root of 136 is more than 11 but less than 12: \( 11 < \sqrt{136} < 12 \) Hence, the numbers that should be in the blank boxes are 11 and 12, read from left to right: \[ 11 < \sqrt{136} < 12 \]
Finding Integers for Square Root of 141
This problem involves finding the two integers that the square root of 141 lies between. The square root of 141 is an irrational number because 141 is not a perfect square; thus, it cannot be expressed as an exact integer. To solve this question, we need to find the two perfect squares nearest to 141, one that is less than 141 and one that is greater than 141. The perfect square closest to 141 but less than 141 is 121, because \( \sqrt{121} = 11 \). The next perfect square greater than 141 is 144, because \( \sqrt{144} = 12 \). Therefore, the square root of 141 is greater than 11 but less than 12. Putting this into the inequality given in the image: 11 < \( \sqrt{141} \) < 12 So the boxes in the inequality would be filled in with 11 and 12, accordingly.
Understanding Number Range Around Square Roots
The question in the image seems to be asking to fill in the blanks with numbers that are less than and greater than the square root of 34, respectively. Let's find two perfect squares that are close to 34 to help us find the square root range of 34: Taking perfect squares near 34: - \(5^2 = 25\) - \(6^2 = 36\) Since 25 is less than 34, and 36 is greater than 34, the square root of 34 will be between 5 and 6. Therefore, a number less than square root of 34 would be 5 (or any number between 5 and \(\sqrt{34}\) that's not an integer), and a number greater than square root of 34 would be 6 (or any number between \(\sqrt{34}\) and 6 that's not an integer). So, the blanks should be filled with: 5 < \(\sqrt{34}\) < 6
Determining Perfect Squares for Square Root Range
The question in the image asks to find two perfect squares between which the square root of 34 falls. Since \(\sqrt{34}\) is not a whole number, we know it lies between two consecutive integer square roots. To determine these, we find the perfect squares nearest to 34 on either side. The perfect square less than 34 is \(5^2 = 25\), and the perfect square greater than 34 is \(6^2 = 36\). Hence, \(\sqrt{25}\) is 5, and \(\sqrt{36}\) is 6. So the square root of 34 falls between the square roots of 25 and 36. Therefore, the inequalities are: \[ 5 < \sqrt{34} < 6 \] In the provided boxes, the numbers 5 and 6 would fit appropriately to show the inequality. \[ 5 < \sqrt{34} < 6 \]
Solving Inequality with Square Root Approximation
The image shows an inequality with a square root of 139, and you need to determine what numbers could go into the blank boxes so that the inequality is true. Firstly, let's find out the approximate value of √139 since it's not a perfect square. To get an idea of where it lies, you can compare it to perfect squares nearby. For instance: - √121 = 11 (since 11^2 = 121) - √144 = 12 (since 12^2 = 144) Since 139 is between 121 and 144, √139 will be between 11 and 12. Calculating the exact decimal would give you a better approximation: √139 ≈ 11.789... This means that we need to find integers that are immediately less than and greater than 11.789. The integer immediately less than 11.789 is 11, and the integer immediately greater than 11.789 is 12. Therefore, to satisfy the inequality, the numbers on the blank boxes should be 11 and 12 in that order: 11 < √139 < 12
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